diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpmrd" "b/data_all_eng_slimpj/shuffled/split2/finalzzpmrd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpmrd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\nError-correction is required whenever information has to be reliably\n transmitted through a noisy environment. The theoretical\n grounds for classical error-correcting codes were first presented in \n1948 by Shannon \\cite{shannon}. He showed that it is possible to transmit \n information trough a noisy channel with a vanishing error probability\n by encoding \n up to a given critical rate $R_c$ equivalent to the \n{\\it channel capacity}.\n However, Shannon's arguments were non-constructive and devising such\n codes turned out to be a major practical problem in the area of information \ntransmission. \n\nIn 1989 Sourlas \\cite{sourlas89,sourlas94} proposed that, due to the\nequivalence between addition over the field $\\{0,1\\}$ and multiplication over\n$\\{{\\pm 1}\\}$, many error-correcting codes can be mapped onto many-body \nspin-glasses with appropriately defined couplings. This observation opened\nthe possibility of applying techniques from statistical physics to \nstudy coding systems, in particular, these ideas were applied to the study of \nparity check codes.\nThese linear block codes can be represented by matrices of $N$ columns\nand $M$ rows that transform $N$-bit messages to $M$ ($>N$) parity checks.\nEach row represents bits involved in a \nparticular check and each column represents checks involving\nthe particular bit. The number of bits used in each check and the number \nof checks per bit depends on the code construction. We concentrate on the \ncase where exactly $C$ checks are performed for each bit and exactly \n$K$ bits compose each check. \n\nThe {\\it code rate} $R$ is defined as the information conveyed per channel \nuse $R=H_2(f_s)N\/M=H_2(f_s)K\/C$, where $H_2(f_s)=\n-(1-f_s)\\;\\mbox {log}_2 (1-f_s)\\;-\n\\;f_s\\;\\mbox {log}_2 (f_s) $ is the binary entropy of the message with bias \n$f_s$. \n\n\\begin{figure}\n\\hspace*{.4cm}\n\\epsfxsize=150mm \\epsfbox{figure1.ps}\n\\vspace{0.5cm}\n\\caption{The encoding, message corruption in the noisy channel and \n decoding can be represented as a Markovian process. The aim is to obtain \na good estimative $\\mbox{\\boldmath $\\widehat {\\xi}$}$ for the \noriginal message $\\xi$.}\n\\label{encode}\n\\end{figure}\n\nIn the mapping proposed by Sourlas a message is represented by a \nbinary vector $\\mbox{\\boldmath $\\xi$}\n\\in\\{\\pm 1\\}^N$ encoded to a higher dimensional vector\n$\\mbox{\\boldmath $J^0$}\\in\\{\\pm 1\\}^M$ defined as $J^{0}_{\\langle i_{1},\ni_{2} \\ldots i_{K}\\rangle} = \\xi_{i_{1}} \\xi_{i_{2}} \\ldots \\xi_{i_{K}}$,\nwhere $M$ sets of $K$ indices are randomly chosen. A corrupted version \n$\\mbox{\\boldmath $J$}$ of the encoded message $\\mbox{\\boldmath $J^0$}$ has to \nbe decoded for retrieving the original message. The decoding process can be \nviewed as a statistical Bayesian process \\cite{iba98} (see Fig.\\ref{encode}). \n Decoding focuses on producing an estimate \n $\\widehat{\\mbox{\\boldmath $\\xi$}}$ to the original message that minimizes a given expected \nloss $\\langle\\langle {\\cal L}(\\xi,\\widehat{\\xi})\\rangle_{p(J\\mid\\xi)}\\rangle_\n{p(\\xi)}$ averaged over the indicated probability distributions. The \ndefinition of the loss depends on the particular task; the simple Hamming \ndistance ${\\cal L}(\\xi,\\widehat{\\xi})=\\sum_j \\xi_j \\widehat{\\xi}_j$ can be \nused for decoding binary messages. An optimal estimator for this particular\nloss function is $\\widehat{\\xi}_j=\\mbox{sign}\\langle S_j \n\\rangle_{p(S\\mid J)}$ \\cite{iba98}, where $\\mbox{\\boldmath $S$}$ \nis a $N$ dimensional binary vector representing outcomes of the \ndecoding process.\n Using Bayes' theorem, the posterior \nprobability can be written as $\\mbox{ln }p(\\mbox{\\boldmath $S$}\\mid\\mbox{\\boldmath $J$} )=\\mbox{ln }p(\\mbox{\\boldmath $J$}\\mid \\mbox{\\boldmath $S$})\n+ \\mbox{ln }p(\\mbox{\\boldmath $S$}) + \\mbox{const}$. \nSourlas has shown \\cite{sourlas94} that for parity check codes this \nposterior can be written as a many-body Hamiltonian:\n\\begin{eqnarray}\n\\label{eq:Hamiltonian} \n\\mbox{ln }p(\\mbox{\\boldmath $S$}\\mid\\mbox{\\boldmath $J$} )&=&-\\beta\\; {\\cal H}(\\mbox{\\boldmath $S$})\\nonumber\\\\\n&=&\\beta \\sum_{\\mu} \n{\\cal A}_{\\mu} \\ J_{\\mu} \\ \\prod_{i\\in\\mu} S_{i} + \n\\beta{\\cal H}_{\\mbox{\\scriptsize prior}} (\\mbox{\\boldmath $S$}),\n\\end{eqnarray}\nwhere $\\mu=\\left\\langle i_{1},\\ldots i_{K} \\right\\rangle$ is a set of indices \nand ${\\cal A}$ is a tensor with the properties ${\\cal A}_\\mu\\in\\{0,1\\}$ and \n $\\sum_{\\mu\\setminus i}{\\cal A}_\\mu=C$ $\\forall i$, which determines the $M$ \ncomponents of the codeword $\\mbox{\\boldmath $J$}^{0}$. The second term ${\\cal H}_{\\mbox{\\scriptsize prior}} (\\mbox{\\boldmath $S$})$ stands for \nthe prior knowledge on the actual messages; it can\nbe chosen as ${\\cal H}_{\\mbox{\\scriptsize prior}}(\\mbox{\\boldmath $S$})=F\\sum_{j=1}^{N} S_j$\n to represent the expected bias in the \nmessage bits. For the simple case of a memoryless binary symmetric channel\n(BSC), $\\mbox{\\boldmath $J$}$ is a corrupted version of the\n transmitted message $\\mbox{\\boldmath $J$}^{0}$ where each bit is \nindependently flipped with probability $p$ during \n transmission. The hyper-parameter $\\beta$, that reaches \nan optimal value at Nishimori's temperature \\cite{iba98,rujan93,nishi},\n is related to the channel corruption rate.\nThe decoding procedure translates to finding the thermodynamical spin \naverages for the system defined by the Hamiltonian (\\ref{eq:Hamiltonian}) at \na certain temperature (Nishimori's temperature for optimal decoding); as the \noriginal message is binary, the retrieved message bits are given by the signs \nof the corresponding averages.\n\nIn the statistical physics framework the performance of the \nerror-correcting process can be measured by the overlap between actual \nmessage and estimate for a given scenario characterized by a code rate, \ncorruption process and information content of the message. To asses\nthe typical properties we average this overlap over all possible\ncodes $\\cal A$ and noise realizations (possible corrupted vectors\n$\\mbox{\\boldmath $J$}$) given the message $\\mbox{\\boldmath $\\xi$}$ and\nthen over all possible messages: \n\n\\begin{equation}\n\\label{eq:mag}\nm=\\frac {1}{N}\\left \\langle \\sum_{i=1}^N {\\xi}_i \\;\\left \\langle \n\\mbox{sign}\\langle S_i \\rangle \\right\\rangle_{{\\cal A},J|\\xi}\\right\n\\rangle_{\\xi}\n\\end{equation}\nHere $\\mbox{sign}\\langle S_i \\rangle$ is the sign of the spins thermal \naverage corresponding to the Bayesian optimal decoding. The average error \nper bit is then given by $p_e = (1-m)\/2 $. Although this performance measure is not the usual physical magnetization (it can be better described as a measure of misalignment of the decoded message), for brevity, \n we will refer to it as {\\it magnetization}.\n\nFrom the statistical physics point of view, the number of checks per bit is \nanalogous to the spin system connectivity and the number of\nbits in each check is analogous to the number of spins per interaction.\nSourlas' code has been studied in the case of extensive connectivity \n, where the\nnumber of bonds $C \\!\\sim\\!$ \\scriptsize $\\left( \\begin{array}{c} N-1 \\\\ K-1\n\\end{array} \\right)$ \\normalsize scales with the system size. In\nthis case it can be mapped onto known problems\nin statistical physics such as the SK \\cite{SK} ($K\\!\\!=\\!\\!2$) and \nRandom Energy (REM) \\cite{Derrida_REM} ($K \\!\\!\\rightarrow\\!\\! \n\\infty$) models. It has been shown that the REM saturates \nShannon's bound \\cite{sourlas89}. However, it has a rather limited practical \nrelevance as the choice of extensive connectivity corresponds to a \nvanishingly small code rate. \n\n\nHere we present an analysis of Sourlas' code for the case of finite\nconnectivity where the code rate is non-vanishing,\ndetailing and extending our previous brief reports \\cite{ks98a,ks98b}.\nWe show that Shannon's bound can also be attained at finite code rates.\nWe study the decoding dynamics and discuss the connections between \nstatistical physics and belief propagation methods. \n\n\n This paper is organized as follows: in Section II we introduce a naive \nmean-field\nmodel that contains all the necessary ingredients to understand the system qualitatively. Section III describes the statistical\n physics treatment of Sourlas' code showing that Shannon's bound\n can be attained for finite code rates if $K\\rightarrow\\infty$. The finite $K$ case and the Gaussian noise are also discussed in Section III. The decoding dynamics is analyzed in Section IV. Concluding remarks are given in \nSection V. Appendices with detailed calculations are also provided.\n\n\n\n\\section{Naive Mean Field Theory}\n\\label{sec:naive} \n\n\\subsection {Equilibrium}\n\n\\begin{figure}\n\\hspace*{.4cm}\n\\epsfxsize=120mm \\epsfbox{figure2.eps}\n\\caption{Code performance measured by the magnetization $m$ as a function \nof the noise level $p$ as given by the naive mean-field theory at code rate \n$R=1\/2$ and $K=2,3,4$ respectively from the bottom. The long-dashed line \nindicates PARA-FERRO coexistence. Insets: Maximum initial \ndeviation $\\lambda$ for convergence at a noise level $p=0.1$. Top inset:\n $K=3$ and increasing $C$. Bottom inset: Code rate $R=1\/2$ and \nincreasing $K$.}\n\\label{naive}\n\\end{figure}\n\n\nTo gain some insight into the code behavior one can start by considering \nthat the original message is $\\xi_j=1$ for all $j$ (so $m=1$ will correspond \nto perfect decoding) and use Weiss' mean-field theory as a first (naive) \napproximation. The idea is to consider an\neffective field given by (for unbiased messages with $F=0$):\n\\begin {equation}\n\\label{eq:naiveeff}\nh^{\\mbox{\\scriptsize eff}}_j =\\sum_{\\{\\mu:j\\in \\mu\\}}J_\\mu \\prod_{i\\in\\mu\\setminus j} S_i\n\\end{equation}\nacting in every site. The first strong approximation here consists in\n disregarding \nthe reaction fields that describe the influence of site $j$ back over the \nsystem.\nThe local magnetization can then be calculated:\n\\begin {equation}\n\\label{eq:naivemag}\nm_j=\\left\\langle\\mbox{tanh}\\left(\\beta h^{\\mbox{\\scriptsize eff}}_j \\right)\\right\\rangle_{J,S}\\simeq \\mbox{tanh}\\beta \\left\\langle h^{\\mbox{\\scriptsize eff}}_j \\right\\rangle_{J,S},\n\\end{equation}\nwhere we introduced a further approximation taking averages inside the \nfunction that can be seen as a high temperature approximation. Disregarding correlations among spins and computing \nthe proper averages one can write:\n\\begin {equation}\n\\label{eq:naivemagav}\nm=\\mbox{tanh} \\left( \\beta \\; C (1-2p)\\; m^{K-1}\\right),\n\\end{equation}\nwhere $p$ is the noise level in the channel. An alternative way to derive the above equation is by considering the free-energy:\n\\begin {equation}\n\\label{eq:naivefree}\nf(m)=-(1-2p)\\frac{C}{K}m^{K}-\\frac{s(m)}{\\beta}.\n\\end{equation}\nThe entropic term $s(m)$ is:\n\\begin {equation} \ns(m)=-\\frac{1+m}{2}\\mbox{ ln}\n\\left(\\frac{1+m}{2}\\right)-\\frac{1-m}{2}\\mbox{ ln}\n\\left(\\frac{1-m}{2}\\right).\n\\end{equation}\nMinimizing this free-energy one can obtain Eq.(\\ref{eq:naivemagav}) whose\nsolutions give the possible phases after the decoding process. In Fig.\n\\ref{naive} we show the maximum magnetization solutions $m$ for \nEq.(\\ref{eq:naivemagav}) as a function of the flip rate $p$ at code rate \n$R=1\/2$ and $K=2,3,4$. For $K=2$ the performance degrades faster with the \nnoise level than in the $K>2$ case. The dashed line indicates coexistence \nbetween paramagnetic (PARA) $m=0$ and ferromagnetic (FERRO) $m>0$ phases. \n\n\n\\subsection{Decoding Dynamics}\n\n\\begin{figure}\n\\hspace*{3cm}\n\\epsfxsize=50mm \\epsfbox{figure3.ps}\n\\vspace{0.5cm}\n\\caption{Graph representing a code.}\n\\label{node}\n\\end{figure}\n\n\n\nIn a naive mean-field framework the decoding process can be \nseen as an iterative solution for (\\ref{eq:naivemagav}) starting from a\nmagnetization value that depends on the prior knowledge about the \noriginal message. The fixed points of this dynamics correspond to the minima of the free-energy; a specific minimum is reached depending on the initial condition. In the insets of Fig.\\ref{naive} we show, as a measure for the basin of attraction, the maximal deviation between\nthe initial condition and the original message $\\lambda=1-m_0$ that allows \nconvergence to a FERRO solution. At the bottom inset we show the deviation \n$\\lambda$ at code rate $R=1\/2$, increasing values of $K$ and \n noise level $p=0.1$ . \nAn increasing initial magnetization is needed when $K$ increases, \ndecoding without prior knowledge is only possible for $K=2$. The top \ninset shows $\\lambda$ for $K=3$, $p=0.1$; as $C$ increases (code rate decreases), the basin of attraction increases. \n\nOne can understand intuitively how the basin of attraction depends on \nthe connectivities by representing the code in a graph with bit and check nodes and looking at the mean-field \nbehavior of a single bit node (see Fig.\\ref{node}). The corrupted checks\n contribute wrong ($-1$ for the ``all ones'' message case) values to the bit nodes ($m<1$ in the mean field). Since check node values correspond to a product of $K-1$ bit values, the probability of updating these nodes to the wrong values increases with $K$, degrading\nthe overall performance. On the other hand, if $C$ increases for a fixed $K$ the bit nodes gather more information and are less sensitive to the presence of (a limited amount of ) wrong bits .\n\n\nAlthough this naive picture indicates some of the qualitative features \nof real codes, one certainly cannot rely in its numerical predictions. \nIn the following sections we will study Sourlas' codes using more \nsophisticated techniques that will substantially refine the analysis. \n\n\n\n\n\\section{Equilibrium}\n\n\n\\subsection{Replica Theory} \n\\label{sec:replica}\n\n\n\nIn the following subsections we will develop the replica symmetric theory for Sourlas' codes and show that, in addition to providing a good description of the equilibrium, it describes the typical decoding dynamics using \n belief propagation methods. \n\nThe previous naive ``all ones'' messages assumption can be formally translated\nto the gauge transformation \\cite{frad} $S_{i} \\!\\!\\mapsto\\!\\!\n S_{i} \\xi_{i}$ and $J_{\\mu}\\!\\!\\mapsto\\!\\! J_{\\mu}\\prod_{i\\in \\mu} \\xi_{i} $ \nthat maps any general message to the FERRO configuration defined as \n$\\xi_i^{*}=1$ $\\forall i$. One can then rewrite the Hamiltonian in the form:\n\\begin{equation}\n\\label{eq:Hamiltonian_gauge} {\\cal H}(\\mbox{\\boldmath $S$})=- \n\\sum_{\\mu} {\\cal A}_{\\mu} \\ J_{\\mu} \\ \\prod_{i\\in\\mu} S_{i} -\nF \\sum_{k}\\xi_k S_{k} \\ ,\n\\end{equation} \n\nWith this transformation, the bits of the uncorrupted encoded message are \n $J^0_i=1$ $\\forall i$ and, for a BSC, the corrupted bits are random variables with probability:\n\\begin{equation}\n\\label{eq:xi_J_prob_dist}\n{\\cal P}\\left(J_{\\mu}\\right) = (1\\!-\\!p)\\ \\delta \\left(J_{\\mu} \\!-\\! 1 \\right) + p \\ \\delta\\left(J_{\\mu} \\!+\\! 1 \\right), \n\\end{equation}\nwhere $p$ is the channel flip rate. For deriving typical properties of these codes one has obtain an expression for the free-energy by invoking the replica approach where the free-energy is defined as:\n\\begin{equation}\n\\label{eq:freenergy}\nf= -\\frac{1}{\\beta}\\lim_{N\\rightarrow\\infty} \\frac{1}{N}\n\\left.\\frac{\\partial} {\\partial {\\mathit n}}\\right |_{{\\mathit n}=0} \\langle {\\cal Z}^{ \\mathit n}\\rangle_{{\\cal A},\\xi,J}, \n\\end{equation}\nwhere $\\langle {\\cal Z}^{ \\mathit n}\\rangle_{{\\cal A},\\xi,J}$ represents\nan analytical continuation in the interval $n\\in[0,1]$ of the replicated\npartition function defined as:\n\n\\begin{equation}\n\\label{eq:partit}\n\\langle {\\cal Z}^{n}\\rangle_{{\\cal A},\\xi,J} = \\mbox{Tr}_{\\{S_j^\\alpha\\}}\n\\left[\\left \\langle e^{ \\beta F \\sum_{\\alpha,k}\\xi_k S^\\alpha_{k}}\\right \n\\rangle_{\\xi}\\left\\langle \\exp\\left(\\beta\\sum_{\\alpha,\\mu} \n{\\cal A}_{\\mu} \\ J_{\\mu} \\ \\prod_{i\\in\\mu} S^\\alpha_{i} \\right) \n\\right\\rangle_{{\\cal A},J} \\right]. \n\\end{equation}\n\n\n The magnetization can be \nrewritten in the gauged variables as :\n\\begin{equation}\n\\label{eq:mag_gauged}\nm= \\left \\langle\\left \\langle \\mbox{sign}\\langle S_i \\rangle \n\\right\\rangle_{{\\cal A},J|\\xi^*}\\right \\rangle_{\\xi},\n\\end{equation}\nwhere $\\xi^*$ denotes the transformation of a message $\\xi$ into \nthe FERRO configuration. The usual magnetization per site can be \neasily obtained by calculating \n\\begin{equation}\n\\label{eq:fundamental}\n\\left \\langle\\left \\langle S_i \\right \\rangle\\right\\rangle_{{\\cal A},J,\\xi}=- \\left( \\frac {\\partial f} { \\partial(\\xi F)} \\right). \n\\end{equation}\nFrom this derivative one can find the distribution of the effective local \nfields $h_j$ that can be used to asses the magnetization $m$, since \n$\\mbox{sign}\\left(\\langle S_j\\rangle\\right)=\\mbox{sign}(h_j)$ .\n\nTo compute the replicated partition function we closely follow \nRef. \\cite{wong_a}. We average uniformly over all codes ${\\cal A}$ such \nthat $\\sum_{\\mu\\setminus i}{\\cal A}_{\\mu}= C$ $\\forall i$ to find: \n\n\\begin{eqnarray}\n\\label{eq:partit_2}\n\\langle {\\cal Z}^{n}\\rangle_{{\\cal A},\\xi,J}& =&\\exp \\left\\{ N \\;Extr_\n{q,\\widehat{q}}\\left[C-\\frac{C}{K}+\\frac{C}{K}\\left(\\sum_{l=0}^{n}{\\cal T}_l\\sum_{\\langle \\alpha_1 \\ldots \\alpha_l\\rangle}\nq_{\\alpha_1 \\ldots \\alpha_l}^{K} \\right)\\right.\\right.\\nonumber\\\\ \n& -&\\left.\\left. C \\left(\\sum_{l=0}^{n}\\sum_{\\langle \\alpha_1 \\ldots\n\\alpha_l\\rangle}q_{\\alpha_1 \\ldots \\alpha_l}\\widehat{q}\n_{\\alpha_1 \\ldots \\alpha_l}\\right) \\nonumber \\right. \\right.\\\\ \n&+& \\left.\\left.\\ln \\mbox{Tr}_{\\{S^{\\alpha}\\}}\\left \\langle e^{\\beta F\\xi\\sum\n_{\\alpha}S^\\alpha}\\right\\rangle_{\\xi}\\left(\\sum_{l=0}^{n} \n\\sum_{\\langle \\alpha_1 \\ldots \\alpha_l\\rangle}\\widehat{q}\n_{\\alpha_1 \\ldots \\alpha_l}S^{\\alpha_1}\\ldots S^{\\alpha_l} \\right)^C \\right]\\right\\}, \n\\end{eqnarray}\nwhere ${\\cal T}_l=\\langle \\tanh^l(\\beta J) \\rangle_J$, as \nin \\cite{viana}, and $q_0=1$. We give details of this calculation in \nthe Appendix A.\nAt the extremum the order parameters acquire expressions similar to those of\n Ref. \\cite{wong_a}:\n\n\\begin{eqnarray}\n\\label{order-param}\n\\widehat{q}_{\\alpha_1,...,\\alpha_l}&=& {\\cal T}_l\\; q^{K-1}_\n{\\alpha_1,...,\\alpha_l}\\nonumber \\\\\nq_{\\alpha_1,...,\\alpha_l}&=&\\left \\langle \\left (\\prod_{i=1}^l\nS^{\\alpha_i} \\right) \\left(\\sum_{l=0}^{n}\\sum_{\\langle \\alpha_1 \\ldots \n\\alpha_l\\rangle}\\widehat{q}_{\\alpha_1 \\ldots \\alpha_l}S^\n{\\alpha_1}\\ldots S^{\\alpha_l}\\right)^{-1}\\right\\rangle_{\\cal X}.\n\\end{eqnarray}\nwhere \n\\begin{equation}\n{\\cal X}=\\left \\langle e^{\\beta F\\xi\\sum_{\\alpha}S^\\alpha}\\right\\rangle_\n{\\xi}\\left(\\sum_{l=0}^{n}\\sum_{\\langle\n\\alpha_1 \\ldots\\alpha_l\\rangle}\\widehat{q}_{\\alpha_1 \\ldots \\alpha_l}\nS^{\\alpha_1}\\ldots S^{\\alpha_l} \\right)^{C},\n\\end{equation}\nand $\\langle...\\rangle_{\\cal X}=\\mbox{Tr}_{\\{S^{\\alpha}\\}}\\left[(...)\n{\\cal X}\\right]\/\\mbox{Tr}_{\\{S^{\\alpha}\\}}\\left[(...)\\right]$.\nThe term \n$\\widehat{p}(\\underline {S})=\\sum_{l=0}^{n}\\sum_{\\langle \\alpha_1 \\ldots\\alpha_l\\rangle}\n\\widehat{q}_{\\alpha_1 \\ldots \\alpha_l} S^{\\alpha_1}\\ldots S^{\\alpha_l}$ \nrepresents a probability distribution \nover the space of replicas and $p_0(\\underline{S})=\\left \\langle e^{\\beta F\\xi\\sum_{\\alpha}S^\\alpha}\\right\\rangle_{\\xi}$ is a prior distribution over the same space. For reasons that will become clear in \nSection \\ref{sec:decoding}, $q_{\\alpha_1,...,\\alpha_l}$ represents one\n $l$-th momentum of the\nequilibrium distribution of a bit-check edge in a belief network during the\n decoding process and $\\widehat{q}_{\\alpha_1 \\ldots \\alpha_l}$ \nrepresents $l$-th moments of a check-bit edge\n equilibrium distribution . The distribution ${\\cal X}$ represents the probability of a certain site (bit node) configuration subjected to exactly $C$\ninteractions and with prior probability given by $p_0$. \n\n\n\n\\subsection{Replica Symmetric Solution}\n\\label{sec:symmetric}\n\nThe replica symmetric (RS) ansatz can be introduced via the auxiliary\nfields $\\pi(x)$ and $\\widehat{\\pi}(y)$ in the following way \n(see also \\cite{wong_a}):\n\n\\begin{eqnarray}\n\\label{eq:auxfields}\n\\widehat{q}_{\\alpha_1 ... \\alpha_l}&=&\\int \\: dy \\; \\widehat{\\pi}(y)\n\\tanh^l(\\beta y) ,\\nonumber\\\\\nq_{\\alpha_1 ... \\alpha_l}&=&\\int \\: dx \\; \\pi(x) \\tanh^l(\\beta x)\n\\end{eqnarray}\nfor $l=1,2,\\ldots$. \n\nPlugging it into the replicated partition function (\\ref{eq:partit_2}), \nperforming the limit $n\\rightarrow 0$ and using Eq.(\\ref{eq:freenergy}) \n(see Appendix \\ref{app:B} for details) one obtains:\n\n\\begin{eqnarray}\n\\label{eq:freesym}\nf&=&-\\frac{1}{\\beta}\\: Extr_{\\pi,\\widehat{\\pi}}\\left \\{\\alpha \\ln \\cosh\n\\beta \\right. \\\\\n&+& \\alpha \\int \\left[\n\\prod_{l=1}^{K} dx_{l} \\ \\pi(x_{l}) \\right] \\left\\langle \\ln \\left[ 1\n+ \\tanh \\beta J \\ \\prod_{j=1}^{K} \\tanh \\beta x_{j} \\right]\n\\right\\rangle_{J} \\nonumber \\\\ \n&-& C \\int dx \\ dy \\ \\pi(x) \\\n\\widehat{\\pi}(y) \\ \\ln \\left[ 1 + \\tanh \\beta x \\ \\tanh \\beta y\n\\right] \\nonumber\\\\\n&-& C \\int dy \\ \\widehat{\\pi}(y) \\ \\ln \\cosh \\beta y \\nonumber\\\\\n &+&\\left. \\int \\left[ \\prod_{l=1}^{C} dy_{l} \\ \\widehat{\\pi}(y_{l})\n\\right] \\left\\langle \\ln \\left[ 2 \\cosh \\beta \\left(\\sum_{j=1}^{C}\ny_{j} + F \\xi \\right) \\right]\\right\\rangle_{\\xi} \\right \\}\\nonumber,\n\\end{eqnarray}\nwhere $\\alpha=C\/K$.\nThe saddle-point equations, obtained by varying\nEq.(\\ref{eq:freesym}) with respect to the probability distributions,\nprovide a set of relations between $\\pi(x)$ and $\\widehat{\\pi}(y)$\n\\begin{eqnarray}\n\\label{eq:saddle_point}\n\\fl\\pi(x) &=& \\int \\left[ \\prod_{l=1}^{C-1} dy_{l} \\ \\widehat{\\pi}(y_{l})\n\\right] \\ \\left\\langle \\delta \\left[ x - \\sum_{j=1}^{C-1} y_{j} - F \\xi\n\\right]\\right\\rangle_{\\xi} \\\\ \n\\fl \\widehat{\\pi}(y) &=& \\int \\left[\n\\prod_{l=1}^{K-1} dx_{l} \\ \\pi(x_{l}) \\right] \\ \\left\\langle \\delta \n\\left[ y -\n\\frac{1}{\\beta} \\tanh^{-1} \\left( \\tanh\\beta J \\ \\prod_{j=1}^{K-1}\n\\tanh \\beta x_{j} \\right) \\right] \\right\\rangle_{J} \\ . \\nonumber\n\\end{eqnarray}\n\nLater we will show that this self-consistent pair of equations can be seen as \na mean-field version for the belief propagation decoding.\nUsing Eq.(\\ref{eq:fundamental}) one finds that the local field distribution is\n :\n\\begin{equation}\n\\label{eq:local_field}\n P(h)=\\int \\left[ \\prod_{l=1}^{C} dy_{l} \\\n\\widehat{\\pi}(y_{l}) \\right] \\ \\left\\langle \\delta \\left[ h -\n\\sum_{j=1}^{C} y_{j} - F \\xi \\right]\\right\\rangle_{\\xi},\n\\end{equation}\nwhere $\\widehat{\\pi}(y)$ is given by the saddle point equations above.\n\n\n\nThe magnetization (\\ref{eq:mag}) can then be calculated using:\n\\begin{equation}\n\\label{eq:mag_sym}\nm = \\int d h \\ \\mbox{sign} (h) \\, P(h).\n\\end{equation}\n\nThe code performance can be assessed by assuming a particular prior distribution for the message bits, \nsolving the saddle-point equations (\\ref{eq:saddle_point}) numerically and \nthen computing the magnetization. \n\nInstabilities in the solution within the space of symmetric replicas can be\nprobed looking at second derivatives of the functional whose extremum\ndefines the free-energy (\\ref{eq:freesym}). The simplest necessary\ncondition for stability is having non-negative second functional derivatives\nin relation to $\\pi(x)$ (and $\\widehat{\\pi}(y)$) :\n \n\\begin{equation}\n\\label{eq:stability}\n\\fl \\frac{1}{\\beta} \\int \\left[\\prod_{l=1}^{K-2} dx_{l} \\ \n\\pi(x_{l}) \\right] \\left\\langle\n \\ln \\left[ 1+ \\tanh \\beta J \\ \\tanh^2 \\beta x\\prod_{j=1}^{K-2} \\tanh\n \\beta x_{j} \\right]\n\\right\\rangle_{J} \\geq 0,\n\\end{equation}\nfor all $x$. \nThe replica symmetric solution is expected to be unstable for \nsufficiently low temperatures (large $\\beta$). For high temperatures\nwe can expand the above expression around small $\\beta$ to find the\nstability condition:\n\\begin{equation}\n\\label{eq:stabhigh}\n\\langle J\\rangle_{J} \\langle x \\rangle_{\\pi}^{K-2}\\geq 0 \\; \n\\end{equation}\nWe expect the average $\\langle x \\rangle_{\\pi}=\\int dx\\, \\pi(x)\\, x$ \nto be zero in PARA phase and positive in FERRO phase,\nsatisfying the stability condition. This result is still generally inconclusive, but provides some evidence that can be examined numerically. In Section \\ref{sec:bound} we will test the stability of our solutions using \n condition (\\ref{eq:stability}).\n\nIn the next sections we restrict our study to the unbiased case ($F=0$),\n which is of practical relevance, since it is always possible to compress a biased message to an unbiased one. \n\n\n\n\\subsection {Case $K\\rightarrow\\infty$, $C=\\alpha K$ }\n\\label{sec:Kinfty}\nFor this case one can obtain solutions to the\nsaddle-point equations for arbitrary temperatures. In the first saddle-point \nequation (\\ref{eq:saddle_point}) one can write:\n\\begin{equation}\n\\label{central_limit}\nx=\\sum_{l=1}^{C-1} y_l \\approx (C-1)\\langle y \\rangle_{\\widehat{\\pi}} = (C-1)\\int dy \\; y\\;\n\\widehat{\\pi}(y). \n\\end{equation} \nIt means that if $\\langle y \\rangle_{\\widehat{\\pi}}=0$ (as it is the in \nPARA and spin glass (SG) phases) \nthen $\\pi(x)$ must be concentrated at $x=0$ implying that \n${\\pi}(x)=\\delta(x)$ and $\\widehat{\\pi}(y)=\\delta(y)$ are \nthe only possible solutions. Moreover, Eq.(\\ref{central_limit})\nimplies that in FERRO phase one can expect \n$x\\approx{\\cal O}(K)$. \n\n\nUsing Eq.(\\ref{central_limit}) and the second saddle-point equation \n(\\ref{eq:saddle_point}) one can find a self-consistent equation for the \nmean-field $\\langle y \\rangle_{\\widehat{\\pi}}$:\n\\begin{equation}\n\\label{mean_field}\n\\langle y \\rangle_{\\widehat{\\pi}} = \\left \\langle\\frac{1}{\\beta}\\, \\mbox{tanh}^{-1}\n\\left[\\mbox{tanh}(\\beta J)\\left(\\mbox{tanh}(\\beta (C-1)\\langle y \\rangle_{\\widehat{\\pi}})\n\\right)^{K-1} \\right]\\right\\rangle_J.\n\\end{equation}\nFor a BSC the above average is over distribution \n(\\ref{eq:xi_J_prob_dist}). Computing the average, using $C=\\alpha K$\nand rescaling the temperature as $\\beta = \\tilde{\\beta} (\\mbox {ln}K)\/K $, \nin the limit $K\\rightarrow\\infty$ one obtains: \n\\begin{equation}\n\\label{mean_field2}\n\\langle y \\rangle_{\\widehat{\\pi}} = (1-2p)\\left[\\mbox{tanh}\n(\\tilde{\\beta}\\alpha\\langle y\\rangle_{\\widehat{\\pi}}\n\\mbox{ ln}(K)) \\right]^{K},\n\\end{equation}\nwhere $p$ is the channel flip probability. \nThe mean-field $\\langle y \\rangle_{\\widehat{\\pi}} = 0 $ is always a solution \nto this equation (either \nPARA or SG); at $\\beta_c =\\mbox{ln}(K) \/( 2\\alpha K(1-2p))$\nan extra non-trivial FERRO solution emerges with \n$\\langle y \\rangle_{\\widehat{\\pi}}=1-2p$. As the connection with\nthe magnetization $m$ is given by Eq. (\\ref{eq:local_field}) and \nEq. (\\ref{eq:mag_sym}); it is not difficult to see that it implies $m=1$ for \nFERRO solution. One remarkable\npoint is that the temperature were the FERRO\nsolution emerges is \n$\\beta_c \\sim {\\cal O}(\\mbox{ln}(K)\/K)$; it means that in a simulated\nannealing process PARA-FERRO barriers emerge quite early for large $K$ \nvalues implying metastability and, consequently, a very slow convergence. \nIt seems to advocate the use of small \n $K$ values in practical applications. This case is analyzed in Section\n \\ref{sec:finite}. For $\\beta>\\beta_c$ both PARA and FERRO solutions exist.\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure4.eps}\n\\caption{Phase diagram in the plane temperature $T$ versus noise level $p$ \nfor $K\\rightarrow\\infty$ and $C=\\alpha K$, with\n $\\alpha=4$. The dotted line indicates Nishimori's temperature $T_N$\n. Full lines represent coexistence. The critical noise level is $p_c$.\n The necessary condition for stability in the FERRO phase is satisfied above \nthe dashed line.}\n\\label{phase}\n\\end{figure}\n\n \nThe FERRO free-energy can be obtained from Eq.(\\ref{eq:freesym}) using Eq.(\\ref{central_limit}), being $f_{\\mbox{\\scriptsize FERRO}}=-\\alpha\n(1-2p)$. The corresponding entropy is $s_{\\mbox{\\scriptsize FERRO}}=0$ \nindicating a single solution. The PARA free-energy is obtained by \nplugging $\\pi(x)=\\delta(x)$ and $\\widehat{\\pi}(y)=\\delta(y)$ into Eq. (\\ref{eq:freesym}):\n\\begin{eqnarray}\n\\label{eq:freepara} \nf_{\\mbox{\\scriptsize PARA}}&=& -\\frac{1}{\\beta}(\\alpha \\mbox{ ln}(\\mbox{cosh }\\beta) + \\mbox{ln }2)\n,\\\\\ns_{\\mbox{\\scriptsize PARA}}&=&\\alpha(\\mbox{ln}(\\mbox{cosh }\\beta) -\\beta\\mbox{ tanh }\\beta) +\n\\mbox{ln }2.\n\\end{eqnarray}\nPARA solutions are unphysical for\n$\\alpha > (\\mbox {ln } 2)\/(\\beta \\mbox { tanh }\\beta - \\mbox{ln ch }\\beta)$, \nsince the corresponding entropy is negative.\nTo complete the phase diagram picture we have to assess the spin-glass free-energy and\nentropy. We have seen in the beginning of this section that replica symmetric SG and PARA solutions consist of the same field distributions for $K\\rightarrow\\infty$, \nimplying unphysical behavior. In order to produce a \nsolution with non-negative entropy one has to break the replica symmetry. \nWe use here a pragmatic way to build this solution, using the \nsimplest one-step replica symmetry breaking known as {\\it frozen spins}.\n \nIt was observed in Ref. \\cite{gross} that for the REM a one-step \nsymmetry breaking scheme gives the exact solution. In this scheme the $n$ \nreplicas' space is divided to groups of $m$ identical solutions. \nIt was shown that an abrupt transition in the order parameter from a unique \nsolution (Edwards-Anderson parameter $q=1$, SG phase) to a completely uncorrelated set of solutions ($q=0$, PARA phase) occurs. \nThis transition takes place at a critical temperature\n$\\beta_g$ that can be found by solving the appropriate saddle-point equations;\n this temperature is given by the root of the replica symmetric entropy ($s_{\\scriptsize RS}=0$) meaning that the RS-RSB transition occurs at the same point as the PARA-SG in this model. The symmetry breaking parameter was found to be \n$m_g=\\beta_g\/\\beta$, indicating that this kind of solution \nis physical only for $\\beta>\\beta_g$, since \n$m_g \\leq 1$ \\cite{parisi}, \nindicating a PARA-SG phase transition. The free-energy can be computed by plugging the order parameters in the effective\nHamiltonian, obtained after averaging over the disorder and taking the proper limits. It shows no dependence on the\ntemperature, since for $\\beta>\\beta_g$ the system is completely\nfrozen in a single configuration.\n\n\n\n\nFor the Sourlas' code, in the regime we are interested in, SG solutions\nto the saddle-point equations are given by $\\pi(x)=\\delta(x)$ and\n$\\widehat{\\pi}(y)=\\delta(y)$. The RSB-SG free-energy that guaranties\ncontinuity in the SG-PARA transition is identical to $f_{\\mbox{\\scriptsize PARA}}$,\nsince the SG and PARA solutions have exactly the same structure, to say: \n\n\\begin{equation}\n\\label{eq:free_rsb_sg}\nf_{\\mbox{\\scriptsize RSB-SG}}=-\\frac{1}{\\beta_g}\\;(\\alpha \\;\\mbox{ln}\\,(\\mbox{cosh }\\,\\beta_g) + \n\\mbox{ln }\\,2),\n\\end{equation}\nwhere $\\beta_g$ is a solution for $s_{\\mbox{\\scriptsize RS-SG}}= \\alpha\\;(\\mbox{ln}\\,(\\mbox{cosh }\\,\\beta) -\\beta\\;\\mbox{tanh }\\,\\beta) +\\mbox{ln }\\,2=0$. \n\n In Fig.\\ref{phase} we show the phase diagram for a given code rate $R$ in\nthe temperature $T$ versus noise level $p$ plane. \n\n\n\n\\subsection{Shannon's Limit}\n\\label{sec:bound}\nShannon's analysis shows that up to a critical code rate $R_c$, which equals \nthe channel capacity, it is possible to recover information with arbitrarily \nsmall error probability for a given noise level. For the BSC :\n\\begin{equation}\n\\label{eq:shannon}\nR_c=\\frac{1}{\\alpha_c}=1+p\\mbox{ log}_2 \\;p + (1-p)\\mbox{ log}_2\\; (1-p).\n\\end{equation}\n \nSourlas' code, in the case where \n$K\\rightarrow\\infty$ and $C \\sim {\\cal O}(N^K)$ can be mapped onto the REM \nand has been shown to be capable of saturating Shannon's bound in the limit \n$R\\rightarrow 0$ \\cite{sourlas89}. In this section we extend the analysis\n to show that Shannon's bound can be attained by Sourlas' \ncode at zero temperature also for \n$K\\rightarrow\\infty$ limit but with connectivity $C=\\alpha K$.\nIn this limit the model is analogous to the diluted REM analyzed by Saakian\n in \\cite {saakian}. \nThe errorless phase is manifested in a FERRO phase with perfect \nalignment ($m=1$) (condition that is only possible for infinite $K$) up \nto a certain critical noise level; a further noise level increase produces \nfrustration leading to a SG phase where\nthe misalignment is maximal ($m=0$). The FERRO-SG transition is \nanalogous to the transition from errorless decoding to decoding with \nerrors described by Shannon. A PARA phase is\nalso present when the transmitted information is insufficient to\nrecover the original message ($R>1$). \n\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure5.eps}\n\\caption{Histogram representing the mean-field distribution\n $\\widehat{\\pi}(y)$ obtained by Monte-carlo integration at low \ntemperature ($\\beta=10$, $K=3$,$C=6$ and $p=0.1$). Dotted lines represent\nsolutions obtained by iterating self-consistent equations both with five peak and three peak ans\\\"atze. Inset: detailed view of the weak regular part arising in the Monte-carlo integration. }\n\\label{fields}\n\\end{figure}\n\nAt zero temperature saddle-point equations (\\ref{eq:saddle_point}) can\nbe rewritten as:\n\n\\begin{eqnarray}\n\\label{eq:sp_infty}\n\\fl \\pi(x) &=& \\int \\left[ \\prod_{l=1}^{C-1} dy_{l} \\ \\widehat{\\pi}(y_{l})\\right]\n \\ \\delta \\left[ x - \\sum_{j=1}^{C-1} y_{j} \\right] \\\\\n\\fl \\widehat{\\pi}(y) &=& \\int \\left[\n\\prod_{l=1}^{K-1} dx_{l} \\ \\pi(x_{l}) \\right] \\ \\left\\langle \\delta \n\\left[ y - \\mbox{sign}(J \\prod_{l=1}^{K-1} x_{l}) \\mbox{min}(\\mid J\\mid,\n ... , \\mid x_{K-1} \\mid)\\right] \\right\\rangle_{J} \\ , \n\\nonumber\n\\end{eqnarray}\n\nThe solutions for these saddle-point equations may, in general, result\nin probability distributions with singular and regular parts. As a\nfirst approximation we choose the simplest self-consistent family\nof solutions which are, since $J=\\pm 1$, given by:\n\\begin{eqnarray}\n\\widehat{\\pi}(y)&=&p_+\\delta(y-1)+p_0\\delta(y)+p_-\\delta(y+1)\\\\\n\\pi(x)&=&\\sum_{l=1-C}^{C-1} T_{[p_{\\pm},p_0;C-1]}(l) \\,\\delta(x-l), \n\\end{eqnarray}\nwith\n\\begin{equation}\nT_{ \\left[ p_{+}, p_{0}, p_{-}; C-1 \\right]} (l) = \\sum^{\\prime}_{ \\{k,h,m\\}} \\frac{(C-1)!}{k! \\ h! \\ m!} \\ p_{+}^{k} \\\np_{0}^{h} \\ p_{-}^{m},\n\\end{equation}\nwhere the prime indicates that $k,h,m$ are such that $k-h=l; \\ k+h+m=C-1$.\nEvidence for this simple ansatz comes from Monte-carlo integration of \nEquation (\\ref{eq:saddle_point}) at very low temperatures, that shows solutions comprising three dominant peaks and a relatively weak regular part. \nInside FERRO and PARA phases a more complex singular solution comprising \nfive peaks $\\widehat{\\pi}(y)=p_{+2}\\delta(y-1)+p_{+}\\delta(y-0.5)+p_0\\delta(y)+p_-\\delta(y+0.5)+p_{-2}\\delta(y+1)$ collapses back to the simpler three peak solution. In Fig.\\ref{fields} we show a typical result of a Monte-carlo integration for the field $\\widehat{\\pi}(y)$. The two peak that emerge by using either the three peak ansatz or the five peak ansatz are shown as dotted lines. In the inset we show the weak regular part of the Monte-carlo solution. \n\n\n Plugging \nthe above ansatz in the saddle-point equations one can write a closed set\nof equations in $p_{\\pm}$ and $p_{0}$ that can be solved numerically \n(see appendix D for details).\n\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure6.eps}\n\\caption{Phase diagram in the plane code rate $R$ versus noise level $p$ \nfor $K\\rightarrow\\infty$ and $C=\\alpha K$ at zero \ntemperature. The FERRO-SG coexistence line corresponds to the Shannon's \nbound.}\n\\label{bound}\n\\end{figure}\n \nThe three peak solution can be of three types: FERRO \n($p_+>p_-$), PARA ($p_0=1$) or SG ($p_-=p_+$). \nComputing free-energies and entropies enables one to construct the phase\n diagram. At zero temperature the PARA free-energy is $f_{\\mbox{\\scriptsize PARA}}=-\\alpha$ \nand the entropy is $s_{\\mbox{\\scriptsize PARA}}=(1-\\alpha)\\mbox{ ln }2$, this phase is physical\nonly for $\\alpha<1$, what is expected since it corresponds exactly to the \nregime where the transmitted information is not sufficient to recover \nthe actual message ($R>1$).\n\nThe FERRO free-energy does not depend on the temperature, having\nthe form $f_{\\mbox{\\scriptsize FERRO}}=-\\alpha(1-2p)$ with entropy $s_{\\mbox {\\scriptsize FERRO}}=0$. One can \nfind the \nFERRO-SG coexistence line that corresponds to the maximum performance of a \nSourlas' code by equating \nEq.(\\ref{eq:free_rsb_sg}) and $f_{\\mbox{\\scriptsize FERRO}}$. \nObserving that $\\beta_g=\\beta_N(p_c)$\n(as seen in Fig.\\ref{phase} ) we found that this transition \ncoincides with Shannon's bound \nEq.(\\ref{eq:shannon}). It is interesting to note that in the large $K$ regime\nboth RS-FERRO and RSB-SG free-energies (for $T0$. \nFor the noiseless\ncase $p=0$ the stability condition is satisfied. The instability of \nFERRO phase opens the possibility that Sourlas' code does not saturate \nShannon's bound, since a correction to the\nFERRO solution could change FERRO-SG transition line. However, it was \nshown in Section \\ref{sec:symmetric} that this instability vanishes for large\ntemperatures, what supports, to some extent, the FERRO-SG line obtained and the saturation \nof Shannon's bound in some region, as long as the temperature is lower than Nishimori's temperature.\nFor finite temperatures the stability condition for FERRO solution\ncan be rewritten as:\n\\begin{equation}\n\\left(1+\\mbox{tanh}(\\beta)\\mbox{tanh}^2(\\beta x)\\right)^{(1-p)}\n\\left(1-\\mbox{tanh}(\\beta)\\mbox{tanh}^2(\\beta x)\\right)^p \\ge 1 \\; \\forall x.\n\\end{equation}\nFor $p=0$ the condition is clearly satisfied. For finite $p$ \na critical temperature above which the stability condition is fulfilled can be found numerically.\nIn Fig.\\ref{phase} we show this critical temperature in the phase diagram;\n one can see that there is a considerable region in which our result that \n Sourlas' code can saturate Shannon's bound is supported. Conclusive evidence to \nthat will be given by simulations presented in Section \\ref{sec:decoding}.\n\n\n\n\\subsection{Finite K Case}\n \\label{sec:finite}\n Although Shannon's bound only can be attained in the limit \n $K\\rightarrow\\infty$, it was shown in the Section \\ref{sec:Kinfty} \n that there are some possible drawbacks, mainly in the decoding of \n messages encoded by large $K$ codes, due to large \nbarriers which are expected to occur between PARA and FERRO\nstates. In this section we consider the finite $K$ case, \nfor which we can solve the RS saddle-point equations (\\ref{eq:saddle_point}) \nfor arbitrary \ntemperatures using Monte-carlo integration. We can also obtain solutions\nfor the zero temperature case using the simple iterative method described in\nSection \\ref{sec:bound}. \n \n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure7.eps}\n\\caption{Top: zero temperature magnetization $m$ as a function of the noise level $p$ for various $K$ values at code rate $R=1\/2$, as\n obtained by the iterative method \n. Notice that the RS theory predicts a transition\n of second order for $K=2$ and first order for $K>2$. Bottom:\nRS-FERRO free-energies (white circles for $K=2$ and from the \nleft: $K\\,=\\,3,4,5$ and $6$) and RSB-SG free-energy (dotted line) as \nfunctions of the noise level $p$. \nThe arrow indicates the region where the RSB-SG phase starts to dominate.\n Inset: a detailed view of the RS-RSB transition region.}\n\\label{kfinite}\n\\end{figure}\n\nAt the top of Fig.\\ref{kfinite} we show the zero temperature \nmagnetization $m$ as a function of the noise level $p$ at \ncode rate $R=1\/2$.\n These curves were obtained by using the \nthree peak ansatz of the Section \\ref{sec:bound}. \nIt can be seen that the transition is of second order for $K=2$ and first \norder for $K>3$ similarly to extensively connected models. The transition\nas described by the RS solution tends to $p=0.5$ as $K$ grows. Note that\nthis does not correspond to perfect retrieval since the RSB spin glass\nphase dominates for $p>p_c$ (see bottom of Fig.\\ref{kfinite}). In the bottom \nfigure we plot RS free-energies and RSB frozen \nspins free-energy, from which we determine the \ncritical probability $p_c$ where the transition occurs (pointed by an arrow).\n After the transition, free-energies for $K=3,4,5$ and $6$ acquire values\nthat are lower than the SG free-energy; nevertheless, the entropy \nis negative and these free-energies are therefore unphysical. It is \nremarkable that this critical value does not change significantly\nfor finite $K$ in comparison to infinite $K$.\nObserve that Shannon's bound\ncannot be attained for finite $K$, since $m=1$ exactly only if \n$K\\rightarrow\\infty$. \n\n\n\nThe $K=2$ model with extensive connectivity (SK) is known to be \nsomewhat special, a full Parisi solution\nis needed to recover the concavity of the free-energy and\nthe Parisi order function has a continuous behavior \\cite{mezard}.\nNo stable solution\nis known for the intensively connected model (Viana-Bray model). \nIn order to check the theoretical result obtained one relies on \nsimulations of the decoding process \nat low temperatures. In Section VIII we show that the simulations are in good \nagreement with the theoretical results.\n\n\n\n\\subsection {Gaussian Noise}\n\\label{sec:gauss}\nUsing the replica symmetric free-energy (\\ref{eq:freesym}) and the \nfrozen spins\nRSB free-energy (\\ref{eq:free_rsb_sg}) one can easily extend the analysis\nto other noise types. The general PARA free-energy and entropy can be written:\n\\begin{eqnarray}\n\\label{eq:fparagen}\nf_{\\mbox{\\scriptsize PARA}}&=&-\\frac{1}{\\beta}\\;\\left(\\alpha\\;\\langle \\mbox{ln}\\,(\\mbox{ch }\\beta J)\\rangle_J +\\mbox {ln }\\,2\\right) \\nonumber \\\\\ns_{\\mbox{\\scriptsize PARA}}&=&\\alpha\\;\\left(\\langle \\mbox{ln}\\,(\\mbox{ch }\\beta J)\\rangle_J - \\beta \\langle J\\;\\mbox{tanh}\\,(\\beta J)\\rangle_J\\right)+\\mbox {ln }2. \n\\end{eqnarray} \nThe SG-RSB free-energy is given by :\n\\begin{equation}\n\\label{eq:fsggen}\nf_{\\mbox{\\scriptsize SG-RSB}}=-\\frac{1}{\\beta_g}\\;\\left(\\alpha\\;\\langle \\mbox{ln }(\\mbox{ch }\\beta_g J)\\rangle_J +\\mbox {ln }2 \\right), \n\\end{equation} \nwith $\\beta_g$ defined as the solution of\n\\begin{equation}\n\\label{eq:entropy}\n\\alpha\\;\\left(\\langle \\mbox{ln }(\\mbox{ch }\\beta_g J)\\rangle_J - \\beta_g \n\\langle J\\;\\mbox{tanh }(\\beta_g J)\\rangle_J\\right)+\\mbox {ln }2 =0 .\n\\end{equation}\n\n\n \nThe FERRO free-energy is in general given by \n$f_{\\mbox{\\scriptsize FERRO}}=-\\alpha\\;\\langle J\\rangle_J=-\\alpha\\;\\langle J \\; \n\\mbox{tanh }(\\beta_N J)\\rangle_J$ (see Appendix \\ref{app:nishifree}).\nThe maximum performance of the code is defined by the critical line :\n\\begin{equation}\n\\label{eq:line}\n\\alpha\\left(\\langle \\mbox{ln}(\\mbox{ch }\\beta_g J)\\rangle_J - \\beta_g \n\\langle J\\;\\mbox{tanh}(\\beta_N J)\\rangle_J\\right)+\\mbox {ln }2 =0,\n\\end{equation}\nobtained by equating free-energies in PARA and FERRO phases. \nComparing this expression with entropy (\\ref{eq:entropy}) it can be\nseen that $\\beta_g=\\beta_N$ at the critical line; the \nsame behavior observed in the BSC case. From Eq.(\\ref{eq:line}) one can \nwrite:\n\\begin{equation}\n\\label{eq:capacity}\nR_c=\\beta_N^2 \\frac{\\partial}{\\partial \\beta}\n\\left[\\frac{1}{\\beta}\\langle \\mbox{log}_2 \n\\mbox{ cosh}(\\beta J)\\rangle_J\\right]_{\\beta=\\beta_N},\n\\end{equation}\nthat can be used to compute the performance of the code for arbitrary \nsymmetric noise. \n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure8.eps}\n\\vspace{1cm}\n\\caption{Critical code rate $R_c$ and channel capacity for a binary Gaussian \nchannel as a function of the signal to noise rate $S\/N$ (solid line). \nSourlas' code saturates Shannon's bound. Channel capacity of the \nunconstrained Gaussian channel (dashed line).}\n\\label{gaussian}\n\\end{figure}\n\nSupposing that the encoded bits can acquire totally unconstrained values \nShannon's bound for Gaussian noise is given by \n$R_c=\\frac{1}{2}\\mbox{ log}_2(1 +S\/N)$, where $S\/N$ is the signal to \nnoise ratio, defined as the ratio of source energy per bit \n(squared amplitude) over the spectral density of the noise (variance). \nIf one constrains the encoded bits to binary values $\\{\\pm 1\\}$ the capacity of a Gaussian channel\nis:\n\\begin{equation}\n\\label{eq:capacity_gauss}\nR_c=\\int dJ\\;P(J\\mid 1) \\mbox{ log}_2 P(J\\mid 1) - \\int dJ \\; P(J)\n\\mbox{ log}_2 P(J), \n\\end{equation}\nwhere $P(J\\mid J^0)=\\frac{1}{\\sqrt{2\\pi \\sigma^2}}\\mbox{ exp}(-\\frac{(J-J^0)^2}{2\\sigma^2})$.\n\nIn Fig.\\ref{gaussian} we show the performance of Sourlas' code\nin a Gaussian channel together with the capacities of the \nunconstrained and binary Gaussian channels. We show that \n $K\\rightarrow\\infty$, $C=\\alpha K$ Sourlas' code\nsaturates Shannon's bound for the binary Gaussian channel as well.\nThe significantly lower performance in the unconstrained \nGaussian channel can be trivially explained by the binary coding\nscheme while signal and noise are allowed to acquire real values.\n\n\\section {Decoding Dynamics}\n\\label{sec:decoding}\n\n\n\\subsection{Belief Propagation}\n\\label{sec:belief}\n\n The decoding process of an error-correcting code relies on\ncomputing averages over the marginal posterior probability \n$P(S_j\\mid \\mbox{\\boldmath $J$})$ for each one \nof the $N$ message bits $S_j$ given the corrupted encoded bits \n$J_\\mu$ (checks), where $\\mu=\\langle i_1 \\ldots i_K \\rangle$ is one of \nthe $M$ sets chosen by the tensor ${\\cal A}_\\mu$. \nThe probabilistic dependencies \nexisting in the code can be represented as a bipartite graph known as a \n{\\it belief network} where nodes in one layer correspond to the $M$ \nchecks $J_\\mu$ while nodes in the other to the $N$ bits $S_j$.\n Each check is connected \nto exactly $K$ bits and each bit is connected exactly to $C$ checks \n(see Fig.\\ref{belief}a).\n\nPearl \\cite{pearl} proposed an iterative algorithm for computation of \nmarginal probabilities in belief networks. These algorithms operate by \nupdating beliefs (conditional probabilities) locally and propagating them. \nGenerally the convergence of these iterations \n depends on the absence of loops in the graph. As can be seen\nin Fig.\\ref{belief}a, networks that define error-correcting\ncodes may include loops and convergence problems may occur. \nRecently it was shown that in some cases Pearl's algorithm works\neven in the presence of loops \\cite{weiss}. \n\n\n\\begin{figure}\n\\hspace*{4cm}\n\\epsfxsize=80mm \\epsfbox{figure9.ps}\n\\vspace{1cm}\n\\caption{(a) Belief network representing an error-correcting code. \nEach bit $S_j$ (white circles) is linked to exactly $C$ checks and \neach check (black circles) $J_{\\mu}$ is linked to exactly $K$ bits. \n(b) Graphical representation of the field $r_{\\mu j}$. The grey box\nrepresents the mean field contribution $\\prod_{l\\in {\\cal L}(\\mu)\\setminus j}q_{\\mu l}$ \nof the other bits on the check $J_\\mu$. (c) Representation of one of the\nfields $q_{\\mu l}$. }\n\\label{belief}\n\\end{figure}\n\n\n\nThe particular use of belief networks as decoding \nalgorithms for error-correcting codes based\non sparse matrices was discussed by MacKay in \\cite{mackay95b}. \nIn this work a loop-free approximation for the graph in Fig.\\ref{belief}a \nwas proposed (see \\cite{pearl} for a general discussion\non such approximations). In fact, it was shown in \\cite{urbanke} that the probability of finite length loops in these graphs vanishes with the system size. \n\nIn this framework the\nnetwork is decomposed in a way to avoid loops and the conditional probabilities\n $q^{(S)}_{\\mu j}$ and $r^{(S)}_{\\mu j}$ are computed.\n The set of bits in a check $\\mu$ is defined as ${\\cal L}(\\mu)$ and the set of checks over the bit $j$ as ${\\cal M}(j)$. The probability that $S_j=S$\n given information on all checks other than $\\mu$ is denoted\n $q^{(S)}_{\\mu j} =P(S_j=S\\mid \\{J_{\\nu}:\\nu \\in {\\cal M}(j)\\setminus\\mu\\})$ \n and $r^{(S)}_{\\mu j} = \\mbox{Tr}_{\\{S_l:l\\in{\\cal L}(\\mu)\\setminus j\\}}\nP(J_{\\mu}\\mid S_j=S, \\{S_l:l \\in {\\cal L}(\\mu)\\setminus j \\}) \n\\prod _{l\\in {\\cal L}(\\mu)\\setminus l} q^{(S_l)}_{\\mu l}$ is the probability of the check $J_{\\mu}$ if the bit $j$ is fixed to $S_j=S$ and the other bits \ninvolved are supposed to have distributions given by $q^{(S_i)}_{\\mu i}$\n. In Fig.\\ref{belief}b one can see a graphical representation of\n$r^{(S)}_{\\mu j}$ that can be interpreted as the influence of the bit $S_j$ and the mean-field $\\prod _{l\\in {\\cal L}(\\mu)\\setminus l} q^{(S_l)}_{\\mu l}$ \n(representing bits in ${\\cal L}(\\mu)$ over than $l$) over the check $J_\\mu$. In the Fig.\\ref{belief}c \nwe see that each field $q^{(S)}_{\\mu l}$ represents the influence of the \nchecks in ${\\cal M}(l)$, excluding $\\mu$, over each bit $S_l$, this \nsetup excludes the loops that may exist in the actual network.\n\nEmploying Bayes theorem, $q^{(S)}_{\\mu j}$ can be rewritten as:\n\\begin{equation}\nq^{(S)}_{\\mu j} = a_{\\mu j}\\;P(\\{J_{\\nu}:\\nu \\in {\\cal M}(j)\\setminus\\mu\\}\\mid S_j)\\;p^{(S)}_{j},\n\\end{equation}\nwhere $a_{\\mu j}$ is a normalization constant such that \n$q^{(+1)}_{\\mu j}+q^{(-1)}_{\\mu j}=1$ and $p^{(S)}_{j}$ is the prior\nprobability over the bit $j$. The distribution $P(\\{J_{\\nu}:\\nu \\in {\\cal M}(j)\\setminus\\mu\\}\\mid S_j)$ can be replaced by a mean-field approximation by \nfactorizing dependencies using fields $r^{(S)}_{\\mu j}$:\n\\begin{eqnarray}\n\\label{eq:belief}\nq^{(S)}_{\\mu j}&=&a_{\\mu j}p^{(S)}_{j}\\prod_{\\nu\\in{\\cal M}(j)\\setminus\\mu}\nr^{(S)}_{\\nu j} \\nonumber \\\\\nr^{(S)}_{\\mu j} &=& \\mbox{Tr}_{\\{S_l:l\\in{\\cal L}(\\mu)\\setminus j\\}}\nP(J_{\\mu}\\mid S_j=S, \\{S_i:i \\in {\\cal L}(\\mu)\\setminus j \\}) \n\\prod _{i\\in {\\cal L}(\\mu)\\setminus j} q^{(S_i)}_{\\mu i}.\n\\end{eqnarray}\n\n\n\n\n\nA message estimate\n$\\widehat\\xi_j =\\mbox{sign}\\left(\\langle S_j \\rangle_{q^{(S)}_j}\\right)$\ncan be obtained by solving the above equations and\ncomputing the pseudo-posterior:\n\\begin{equation}\nq^{(S)}_{j}=a_{j}p^{(S)}_j\\prod_{\\nu\\in{\\cal M}(j)}r^{(S)}_{\\nu j},\n\\label{eq:pseudo}\n\\end{equation}\nwhere $a_{j}$ is a normalization constant. \n \nBy taking advantage of the normalization conditions for the distributions\n$q^{(+1)}_{\\mu j}+q^{(-1)}_{\\mu j}=1$ and \n$r^{(+1)}_{\\mu j}+r^{(-1)}_{\\mu j}=1$\none can change variables and reduce the number of equations (\\ref{eq:belief}) to the couple\n$\\delta q_{\\mu j} = q^{(+1)}_{\\mu j}-q^{(-1)}_{\\mu j}$ and \n$\\delta r_{\\mu j} = r^{(+1)}_{\\mu j}-r^{(-1)}_{\\mu j}$.\nSolving these equations, one can find back $r^{(S)}_{\\mu j}=\\frac{1}{2}\n\\left(1\\;+\\;\\delta r_{\\mu j}S_j \\right)$ and the\npseudo-posterior can be calculated to obtain the estimate.\n\n\n\\subsection{Connection with Statistical Physics}\n\\label{sec:connection}\n\n\nThe belief propagation algorithm was shown in \\cite{mackay95b} to outperform \nother methods such as simulated annealing. In \\cite{ks98a} it was proposed\nthat this framework can be reinterpreted using statistical physics. \nThe main ideas behind the approximations contained in (\\ref {eq:belief}) are somewhat similar to the Bethe \\cite{bethe} approximation to diluted \ntwo-body spin glasses. Actually, \nfor systems involving two-body interactions it is known that the Bethe\napproximation is equivalent to solving exactly a model defined on a \nCayley tree and that this is a good approximation\nfor finitely connected systems in the thermodynamical limit \\cite{wong_d}.\nIn fact, loops in the connections become rare as the system size grows \nand can be neglected without introducing significant errors. The belief propagation can be seen as a Bethe-like approximation for multiple bodies interaction systems.\n\nThe mean-field approximations used here are also quite similar \nto the TAP approach \\cite{tap}. The fields $q^{(S)}_{\\mu j}$ correspond\nto the mean influence of other sites other the site $j$ and \nthe fields $r^{(S)}_{\\nu j}$ represent the influence of $j$ back over the system (reaction fields). \n\nThe analogy can be exposed by observing that the likelihood \n$p(J_\\mu\\mid \\mbox{\\boldmath $S$})$ \nis proportional to the Boltzmann weight:\n\\begin{equation}\nw_B(J_\\mu \\mid \\{S_j:j \\in {\\cal L}(\\mu) \\}) = \\mbox{exp}\\left(-\\beta J_\\mu \\; \\prod_{i\\in\\mu} S_{i}\\right).\n\\end{equation} \nThat can be also written in the more convenient form:\n\\begin{equation}\n\\label{eq:likelihood}\nw_B(J_\\mu \\mid \\{S_j :j \\in {\\cal L}(\\mu) \\}) = \\frac{1}{2} \\mbox{cosh}(\\beta J_\\mu)\\left( 1\\; + \n\\;\\mbox{tanh}(\\beta J_{\\mu})\\; \\prod_{j\\in{\\cal L}(\\mu)}S_j \\right).\n\\end{equation} \n\nThe variable $r_{\\mu j}^{(S_j)}$ can then be seen as proportional to the effective Boltzmann weight obtained by fixing the bit $S_j$:\n\\begin{equation}\n\\label{eq:effective}\nw_{\\mbox{\\scriptsize eff}}(J_\\mu \\mid S_j) \n= \\mbox{Tr}_{\\{S_l\\; :\\;l \\in {\\cal L}(\\mu)\\setminus j \\}}\\; \nw_B(J_\\mu \\mid \\{S_l\\; :\\;l \\in {\\cal L}(\\mu) \\})\\prod _{l\\in {\\cal L}(\\mu)\\setminus j} \n q^{(S_l)}_{\\mu l}. \n\\end{equation} \n Plugging Eq.(\\ref{eq:likelihood}) for the likelihood in equations\n(\\ref{eq:belief}), using the fact that the prior probability is given by $p_j^{(S)}=\\frac{1}{2}\\left(1+\\mbox{tanh}(\\beta S F)\\right)$ and computing $\\delta q_{\\mu j}$ and $\\delta r_{\\mu j}$:\n\n\\begin{eqnarray}\n\\label{eq:tap}\n\\delta r_{\\mu j}&=&\\mbox{tanh}(\\beta J_\\mu) \\prod_{l\\in{\\cal L}(\\mu)\\setminus j} \\delta q_{\\mu l}\\nonumber \\\\\n\\delta q_{\\mu j}&=&\\mbox{tanh}\\left(\\sum_{\\nu\\in{\\cal M}(l)\\setminus \\mu} \n\\mbox{tanh}^{-1}( \\delta r_{\\nu j}) +\\beta F \\right).\n\\end{eqnarray}\n\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure10.eps}\n\\caption{Magnetization as a function of the flip probability $p$ for \ndecoding using TAP equations for $K=2$. From the bottom: \nMonte-carlo solution of the RS saddle-point equations \n for unbiased messages ($f_s=0.5$) at $T=0.26$ (line) and \n$10$ independent runs of TAP decoding for each flip probability (plus signs),\n $T=0.26$ and biased messages ($f_s=0.1$) at \n Nishimori's temperature $T_N$.}\n\\label{TAP1}\n\\end{figure}\n\nThe pseudo-posterior can then be calculated:\n\\begin{equation}\n\\label{eq:pseudoposterior}\n\\delta q_{j}=\\mbox{tanh}\\left(\\sum_{\\nu\\in{\\cal M}(l)}\n\\mbox{tanh}^{-1}( \\delta r_{\\nu j}) +\\beta F \\right),\n\\end{equation}\nproviding Bayes' optimal decoding \n$\\widehat{\\xi}_j=\\mbox{sign}(\\delta q_{j})$. It is \nimportant at this point to support the mean-field \nassumptions used here by methods of statistical \nphysics \\cite{ks98a}. The factorizability of the probability \ndistributions can be explained by weak correlations between connections \n(checks) and by the cluster property: \n\\begin{equation}\n\\lim_{N\\rightarrow \\infty} \\frac{1}{N^2}\\sum_{i\\neq j}\n\\left(\\langle S_i S_j\\rangle_{p(S\\mid J)} -\n\\langle S_i \\rangle_{p(S\\mid J)}\n\\langle S_j \\rangle_{p(S\\mid J)}\\right)^2 \\rightarrow 0\n\\end{equation}\nthat bits $S_j$ obey within a pure state \\cite{mezard}. \n\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure11.eps}\n\\caption{Magnetization as a function of the flip probability $p$ for \ndecoding using TAP equations for $K=5$. \nThe dotted line is the replica symmetric saddle-point equations Monte-carlo\nintegration for unbiased messages ($f_s=0.5$) at the Nishimori's temperature $T_N$. The \nbottom error bars correspond to $10$ simulations using the TAP decoding. \nThe decoding performs badly on average in this scenario. The upper curves\nare for biased messages ($f_s=0.1$) at the Nishimori's temperature $T_N$. \nThe simulations agree with results obtained using the replica symmetric ansatz\nand Monte-carlo integration.}\n\\label{TAP2}\n\\end{figure}\n\n\n\n\nOne can push the above connections even further. Eqs.(\\ref{eq:tap}), of course, depend on the particular received message $\\mbox{\\boldmath $J$}$. In order to make the analysis message independent, one can use \na gauge transformation $\\delta r_{\\mu j} \\mapsto \\xi_j \\delta r_{\\mu j}$ and\n$\\delta q_{\\mu j} \\mapsto \\xi_j \\delta q_{\\mu j}$ to write:\n\\begin{eqnarray}\n\\label{eq:gaugetap}\n\\delta r_{\\mu j}&=&\\mbox{tanh}(\\beta J) \\prod_{l\\in{\\cal L}(\\mu)\\setminus j} \\delta q_{\\mu l}\\nonumber \\\\\n\\delta q_{\\mu j}&=&\\mbox{tanh}\\left(\\sum_{\\nu\\in{\\cal M}(l)\\setminus \\mu} \n\\mbox{tanh}^{-1}( \\delta r_{\\nu j}) +\\beta \\xi_j F \\right).\n\\end{eqnarray}\nIn this form a success in the decoding process correspond to $\\delta r_{\\mu j}>0$ and $\\delta q_{\\mu j}=1$ for all $\\mu$ and $j$. For a large number of iterations, one can expect the ensemble of belief networks to converge to an equilibrium distribution where $\\delta r$ and $\\delta q$ are random variables sampled from distributions $\\widehat{\\rho}(y)$ and $\\rho(x)$ respectively.\nBy transforming these variables as $\\delta r=\\mbox{tanh}(\\beta y)$ and \n$\\delta q=\\mbox{tanh}(\\beta x)$ and considering the actual message and noise as quenched disorder, Eqs.(\\ref{eq:gaugetap}) can be rewritten as:\n\\begin{eqnarray}\n\\label{eq:newgaugetap}\ny&=&\\frac{1}{\\beta}\\left\\langle \\mbox{tanh}^{-1}\\left(\\mbox{tanh}(\\beta J) \\prod_{j=1}^{K-1} \\mbox{tanh}(\\beta x_j)\\right)\\right\\rangle_J\\nonumber \\\\\nx&=&\\left\\langle \\sum_{j=1}^{C-1} y_j + \\xi F\\right\\rangle_\\xi.\n\\end{eqnarray}\n\nThe above relations lead to a dynamics on the distributions $\\widehat{\\rho}(y)$ and $\\rho(x)$, that is exactly the same obtained when solving iteratively \n RS saddle-point equations (\\ref{eq:saddle_point}). The probability distributions $\\widehat{\\rho}(y)$ and $\\rho(x)$ can be ,therefore, identified with $\\widehat{\\pi}(y)$ and $\\pi(x)$ respectively and the RS solutions correspond to decoding a generic message using belief propagation averaged over an ensemble of different codes, noise and signals.\n\n\n\n\nEqs.(\\ref{eq:tap}) are now used to show the agreement between the simulated \ndecoding and analytical calculations. \nFor each run, a fixed\ncode is used to generate 20000 bit codewords from 10000 bit messages,\ncorrupted versions of the codewords are then decoded using (\\ref{eq:tap}).\nNumerical solutions for 10 individual \nruns are presented in Figs.\\ref{TAP1} and \\ref{TAP2},\ninitial conditions are chosen as $\\delta r_{\\mu l}=0$\n and $\\delta q_{\\mu l}=\\mbox{tanh}(\\beta F)$ reflecting prior \n beliefs. In Fig.\\ref{TAP1} \nwe show results for $K=2$ and $C=4$\nin the unbiased case, at code rate $R=1\/2$\n(prior probability $p_j^{(1)}=f=0.5$)\n at a low temperature $T=0.26$ (we avoided $T=0$ due to numerical \ndifficulties). \nSolving saddle-point equations (\\ref{eq:saddle_point}) numerically using \nMonte-carlo integration methods we obtain solutions with good agreement to \nsimulated decoding. In the same figure we show the performance for the case of \nbiased messages ($p_j^{(1)}=f_s=0.1$), at code rate $R=1\/4$. Also here the \nagreement with Monte-carlo integrations is rather convincing. The third \ncurve in Fig.\\ref{TAP1} shows the performance \nfor biased messages at Nishimori's temperature $T_N$, as expected, it\nis far superior compared to low temperature performance and the \nagreement with Monte-carlo results is even better.\n\n\n\n\nIn Fig.\\ref{TAP2} we show the results obtained for $K=5$ and $C=10$.\nFor unbiased messages the system is extremely\nsensitive to the choice of initial conditions and does not perform\nwell in average even at Nishimori's temperature. For biased messages\n($f_s=0.1$, $R=1\/4$) results are far better and in agreement\nwith Monte-carlo integration of the RS saddle-point equations. \n\n\n\n\nThe experiments show that belief propagation methods may be used \nsuccessfully for decoding Sourlas-type codes in practice, and provide \nsolutions that are well described by RS analytical solutions. \n\n\n\n\n\\subsection{Basin of Attraction}\n\\label{sec:basin}\n\n\\begin{figure}\n\\hspace*{2cm}\n\\epsfxsize=120mm \\epsfbox{figure12.eps}\n\\caption{Top: Maximum initial deviation $\\lambda$ for decoding. Top: $\\lambda$\nas function of the number of interactions $K$. Circles are \naverages over $10$ different codes with $N=300$, $R=1\/3$ and noise level \n$p=0.1$. Bottom: $\\lambda$ as function of the connectivity $C$. \nCircles are averages over $10$ codes with $N=300$, $K=3$ \nand noise level $p=0.1$. Lines and $\\times$'s correspond to the RS dynamics \ndescribed by the saddle-point equations. }\n\\label{basins}\n\\end{figure}\n\n\nTo asses the size of the basin of attraction we consider the decoding process as a dynamics in the graphs space where edges\n$\\delta q_{\\mu j}$ are considered as \ndynamical variables. In gauged transformed equations (\\ref{eq:gaugetap})\n, the perfect decoding of a message correspond to $\\delta q_{\\mu j}=1$ .\n To analyse the basin of attraction we start with random initial values \nwith a given normalized deviation from the perfect decoding $\\lambda=\\frac{1}{NC}\\sum_{\\mu j}(1-\\delta q_{\\mu j}^{0})$. It is analogous to the finite magnetizations used in the naive mean-field of Section II, since a given $\\delta q_{\\mu j}^{0}$ corresponds to a given magnetization value by using Eq.(\\ref{eq:pseudoposterior}).\n\nIn Fig.\\ref{basins} we show the maximal deviation in initial conditions required for successful decoding. \nTop figure shows an average over $10$ different codes with\n $N=300$ (circles) for a fixed code rate $R=1\/3$, fixed noise level $p=0.1$ and\nincreasing $K$. Bottom figure shows the maximal deviation in initial conditions for a fixed number of spins per interaction $K=3$, noise level $p=0.1$ and increasing $C$. We confirm the fidelity of the RS description by comparing the experimental results with the basin of attraction predicted by saddle-point equations (\\ref{eq:saddle_point}). One can interpret these\nequations as a dynamics in the space of distributions $\\pi(x)$.\nPerforming the transformation $X=\\mbox{tanh}(\\beta x)$, one can move to\nthe space of distributions $\\Pi(X)$ with support over $[-1,+1]$.\n The initial conditions can then be described simply as $\\Pi^0(X)=(1-\\frac{\\lambda}{2})\\delta(X-1)+\\frac{\\lambda}{2}\\delta(X+1)$. In Fig.\\ref{basins}\nwe show the basin of attraction of this dynamics as lines and $\\times$'s. \n\nThe $K=2$ case is the only practical code from a dynamical point \nof view, since it has the largest basin of attraction and no prior knowledge on the message is necessary for decoding. Nevertheless, this code's performance degrades faster than the $K>2$ case as shown in Section III, which points to a compromise between good dynamical properties in one side and good performance in the other. One idea could be having a code with changing $K$, starting with $K=2$ to guarantee convergence and progressively increasing its values to improve \nthe performance \\cite{idosaad}.\n\n\nOn the other hand, the basin of attraction increases with $C$. Again it \npoints to a trade off between good equilibrium properties \n(small $C$ and large code rates) and good dynamical properties (large $C$, \nlarge basin of attraction). Mixing small and large $C$ values in the same code \nseems to be a way to take advantage of this trade-off \\cite{LMSS,davey,VSK}.\n\n\n\\section{Concluding Remarks}\n\nIn this paper we studied, using the replica\napproach, a finite connectivity many-body spin glass that\ncorresponds to Sourlas' codes for finite code rates. We have shown, \nusing a simplified one step RSB solution for \nspin glass phase, that for $K\\rightarrow\\infty$ and $C=\\alpha K$ \nregime at low temperatures the system exhibits a FERRO-SG phase transition \nthat corresponds to Shannon's bound. However, we have also shown that the\ndecoding problem for large $K$ has bad convergence properties\nwhen simulated annealing strategies are used. \n\nWe were able to find replica symmetric solutions for finite $K$ and \nfound good agreement with practical decoding performance using belief\nnetworks. Moreover, we have shown that RS saddle-point equations actually \ndescribe the mean behavior of belief propagation algorithms.\n\nWe studied the dynamical properties of belief propagation and compared to \n statistical physics predictions, confirming the validity of the description. The basin of attraction was shown to depend on $K$ and $C$. Strategies for improving the performance were discussed.\n\n The same methodology has been recently employed successfully \\cite{kms99}\nto state-of-the-art algorithms as the recent rediscovered Gallager codes\n\\cite{mackay95} and its variations \\cite{idosaad,VSK}.\n We believe that the connections found between belief networks and\nstatistical physics can be further developed to provide deeper insights into\nthe typical performance of general error-correcting codes.\n\n \n\n\n\\label{sec:conclusions}\n\n\\ack\nThis work was partially supported by the program \n``Research For The Future'' (RFTF) of the Japanese Society for the \nPromotion of Science (YK) and by EPSRC grant GR\/L52093 and a Royal Society travel grant (DS and RV). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{s:intro}\nFor a finitely generated {\\it Coxeter group} $(W,S)$ one defines\nthe {\\it Poincar\\'e series} by\n\\begin{equation}\n\\label{eq:poinser}\np_{(W,S)}(t)=\\sum_{w\\in W} t^{\\ell(w)}\\in\\mathbb{Z}\\llbracket t\\rrbracket,\n\\end{equation}\nwhere $\\ell\\colon W\\longrightarrow\\mathbb{N}_0$ denotes the {\\it length function} associated with $(W,S)$.\nIt is well known that $p_{(W,S)}$ is a rational function in $t$\n(cf.~\\cite[Chap.~IV, \\S1, Ex.~25 and 26]{bourbaki--gal46}). \nThis function is explicitly known for finite\nCoxeter groups, and explicitly computable \nfor any given infinite, finitely generated Coxeter group $(W,S)$ using the recursive\nsum formula \n\\begin{equation}\n\\label{eq:altsfor}\n\\frac{1}{p_{(W,S)}(t)}=\\sum_{I\\subsetneq S} (-1)^{|S|-|I|-1}\\frac{1}{p_{(W_I,I)}(t)}\n\\end{equation}\n(cf. \\cite[\\S 5.12]{humphreys--rgcg}). The formula \\eqref{eq:altsfor} suggests that \none should be able to interpret ${p_{(W,S)}(t)}^{-1}$ as an {\\it Euler characteristic},\nbut it is not clear in which context. This point of view\nis also supported by a result of J-P.~Serre who showed that $p_{(W,S)}(1)^{-1}$ coincides\nwith the Euler characteristic of the Coxeter group $W$ (cf. \\cite[\\S 1.9]{serre--cgd}).\nThe main goal of this paper is to show that\n${p_{(W,S)}(q)}^{-1}$ coincides with the Euler characteristic\nof the {\\it Hecke algebra} $\\ca{H}_q(W,S)$ for suitable values of $q$.\n\nLet $R$ be a commutative ring with unit. For certain\naugmented, associative $R$-algebras (cf. \\S\\ref{ss:eueuc})\none can define an Euler characteristic. \nThese algebras will be called {\\it Euler algebras} (cf. \\S\\ref{ss:eaec}).\nFor any distinguished element $q\\in R$\none may define the $R$-Hecke algebra $\\ca{H}=\\ca{H}_q(W,S)$ associated with\nthe Coxeter group $(W,S)$. This algebra can be seen\nas a deformation of the $R$-group algebra\nof the Coxeter group $(W,S)$. It particular, it comes equipped\nwith an \\emph{antipodal map} $\\argu^\\natural\\colon \\ca{H}\\longrightarrow\\ca{H}^{\\op}$, an \n\\emph{augmentation} $\\varepsilon_q\\colon\\ca{H}\\longrightarrow R$, and an $R$-basis \n$\\mathcal{B}=\\{\\,T_{w}\\mid{w}\\in W\\,\\}$.\nMoreover, one has the following.\n\n\\begin{thmA}\nLet $(W,S)$ be a finitely generated Coxeter group, let $R$ be a commutative\nring with unit, and let $q\\in R$ be such that $p_{(W_I,I)}(q)$ is invertible in $R$\nfor any spherical parabolic subgroup $(W_I,I)$.\nThen $\\ca{H}=\\ca{H}_q(W,S)$ is an $R$-Euler algebra, \nand its Euler characteristic is given by $\\chi_\\ca{H}=p_{(W,S)}(q)^{-1}$.\n\\end{thmA}\n\nThe proof of Theorem~A will be established in two steps.\nFirst one has to show that\n$\\ca{H}=(\\ca{H},\\argu^\\natural,\\varepsilon_q,\\mathcal{B})$ is an $R$-Euler algebra;\nthen one has to compute the Hattori-Stalling rank\n$r_{R_q}$, where $R_q$ denotes the left $\\ca{H}$-module associated to $\\varepsilon_q$.\nFor both these goals we will make use of a chain complex $C=(C_\\bullet,\\partial_\\bullet)$\nof left $\\ca{H}$-modules first established by V.V.~Deodhar in\n\\cite{deodhar--sgabo2}. \n\nFor the ring $R=\\mathbb{Z}\\llbracket q\\rrbracket$ the Poincar\\'e series of $(W,S)$ can be written as\n\\begin{equation}\n\\label{eq:poinalt}\np_{(W,S)}(q)=p_{\\ca{H}}=\\sum_{T_w\\in\\mathcal{B}} \\varepsilon_q(T_w)\\in\\mathbb{Z}\\llbracket q\\rrbracket.\n\\end{equation}\nHence one has the identity $p_{\\ca{H}}\\cdot\\chi_{\\ca{H}}=1$ in $\\mathbb{Z}\\llbracket q\\rrbracket$.\nA similar identity involving combinatorial data\nand cohomological data is known for a Koszul algebra $A_\\bullet$ over a field $\\mathbb{F}$;\none has $h_{\\mathbf{A}_\\bullet}(t)\\cdot h_{H^\\bullet(\\mathbf{A}_\\bullet)}(-t)=1$ in $\\mathbb{Z}\\llbracket t\\rrbracket$,\nwhere $h_{\\argu}(t)$ denotes the Hilbert series associated with a \nconnected, graded $\\mathbb{F}$-algebra of finite type (cf. \\cite[p.22, Cor.~2.2]{polishchuk-positselski--qa}).\nIt would be interesting to know whether there exist other examples of\n$\\mathbb{Z}\\llbracket q\\rrbracket$-Euler algebras $\\mathbf{A}$ for which \n$p_{\\mathbf{A}}$ given by \\eqref{eq:poinalt} \nis defined and which satisfy the identity $p_{\\mathbf{A}}\\cdot\\chi_{\\mathbf{A}}=1$.\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank F.~Brenti for some very helpful discussions,\nand M.~Solleveld for pointing out that in a \nslightly different context projective resolutions of affine Hecke algebras were already\nconstructed in~\\cite{opdam-solleveld--haaha}.\nIn an earlier version of this paper the complex $C$ and its properties were\ndiscussed in detail. Our gratitude goes to A.~Mathas and S.~Schroll \nfor informing us that such a complex of left $\\ca{H}$-modules was already \nknown in the literature (cf.~\\cite{deodhar--sgabo2}, \\cite{mathas--qacc})\nand that it is related to a complex of $(\\ca{H},\\ca{H})$-bimodules established in \n\\cite{linckelmann-schroll--tsqacc}.\n\n\n\n\\section{Coxeter groups and Hecke algebras}\n\\label{s:coxhec}\n\n\\subsection{Coxeter groups}\n\\label{ss:cox}\nA \\emph{Coxeter graph} $\\Gamma$ is a finite combinatorial graph%\n\\footnote{In this context the graph $\\emptyset$ with empty vertex \n set is also considered as a Coxeter graph.} \nwith non-oriented edges $\\mathfrak{e}$ labelled by positive integers \n$m(\\mathfrak{e})\\geq 3$ or infinity.\nThe \\emph{Coxeter group} $(W,S)$ associated to $\\Gamma$ consists of the \ngroup $W$ generated by the \nset of involutions $S=\\{\\,s_v\\mid v\\in\\mathfrak{V}(\\Gamma)\\,\\}$ subject to the \nrelations $(s_vs_w)^{m(\\mathfrak{e})}=1$, where $\\mathfrak{e}=\\{v,w\\}\\in\\mathfrak{E}(\\Gamma)$ is an \nedge of label $m(\\mathfrak{e})<\\infty$, and the commutation relations $s_vs_w=s_ws_v$\nwhenever $\\{v,w\\}\\not\\in \\mathfrak{E}(\\Gamma)$.\nThe \\emph{length function} on $W$ with respect to $S$ will be denoted \nby $\\ell\\colon W\\longrightarrow\\mathbb{N}_0$. Since $S=S^{-1}$ is a set of involutions, \n$\\ell(w)=\\ell(w^{-1})$, and it is well known that a longest element $w_0\\in W$\nexists if, and only if, $W$ is finite. In this case it is unique and \nhas the property that $\\ell(w_0 x)=\\ell(w_0)-\\ell(x)$ for all $x\\in W$.\nA Coxeter group which is finite is called \\emph{spherical}, and\n\\emph{non-spherical} otherwise.\nFor a subset $I \\subseteq S$ let $W_I$ be the corresponding parabolic \nsubgroup, i.e., $W_I$ is the subgroup of $W$ \ngenerated by $I$. It is isomorphic to the Coxeter group associated \nto the Coxeter subgraph $\\Gamma^\\prime$ based on the vertices \n$\\{\\,v\\in\\mathfrak{V}(\\Gamma)\\mid s_v\\in I\\,\\}$.\nThe length function of $W$ restricted to $W_I$ coincides with the \nintrinsic length function of the Coxeter group $(W_I,I)$. Put\n\\begin{equation}\n \\label{eq:lcosetreps}\n W^I=\\{\\, w\\in W\\mid \\ell(ws)>\\ell(w) \\text{ for all }s\\in I\\,\\},\n\\end{equation}\nand let ${}^IW=(W^I)^{-1}$, i.e.,\n\\begin{equation}\n \\label{eq:rcosetreps}\n {}^IW=\\{\\, w\\in W\\mid \\ell(sw)>\\ell(w) \\text{ for all }s\\in I\\,\\}.\n\\end{equation}\nOne has the following properties (cf. \\cite[\\S5.12]{humphreys--rgcg}).\n\n\\begin{prop}\n \\label{prop:cosreps}\n Let $(W,S)$ be a Coxeter group, let $w\\in W$ and let $I\\subseteq S$.\n \\begin{itemize} \n \\item[(a)] There exist a unique element $w_I\\in W_I$ and a unique element \n $w^I\\in W^I$ such that $w=w^Iw_I$. Moreover, $\\ell(w)=\\ell(w^I)+\\ell(w_I)$.\n \\item[(b)] There exist a unique element ${}_Iw\\in W_I$ and a unique element\n ${}^Iw\\in {}^IW$ such that $w={}_Iw\\,{}^Iw$. \n Moreover, $\\ell(w)=\\ell({}_Iw)+\\ell({}^Iw)$.\n \\item[(c)] $W^I$ and ${}^IW$ are sets of coset representatives, \n distinguished in the sense that the decomposition is length-additive. \n \\item[(d)] The element $w^I\\in W^I$ is the unique shortest element in $wW_I$. \n \\item[(e)] Let $y\\in W^I$ and $u\\in W_I$. Then $(yu)^I= y$, $(yu)_I=u$, and \n $\\ell(yu)=\\ell(y)+\\ell(u)$.\n \\item[(f)] For $s\\in S$ one has \n $W={}^{\\{s\\}}W \\sqcup s({}^{\\{s\\}}W)$, where $\\sqcup$ denotes disjoint union.\n \\item[(g)] Let $I\\subseteq J\\subseteq S$. Then \n $W^J\\subseteq W^I$. Moreover, $W^S=\\{1\\}$ and $W^\\emptyset=W$. \n \\end{itemize}\n\\end{prop}\n\n\\subsection{Hecke algebras}\n\\label{ss:hec}\nLet $R$ be a commutative ring with unit and with a distinguished element \n$q\\in R$.\\footnote{For certain types it is also possible to consider \n multiple parameter Hecke algebras. \n This is discussed in \\cite{terragni--haacg}.}\nThe \\emph{$R$-Hecke algebra} $\\ca{H}=\\ca{H}_q(W,S)$ associated to $(W,S)$ and $q$\nis the unique associative $R$-algebra which is a free $R$-module with basis\n$\\{\\,T_{w}\\mid {w}\\in W\\,\\}$ subject to the relations\n\\begin{equation}\n \\label{eq:hecrel}\n T_sT_{w}=\n \\begin{cases}\n T_{s{w}}&\\qquad\\text{if $\\ell(s{w})>\\ell({w})$}\\\\\n (q-1)T_{w}+qT_{s{w}}&\\qquad\\text{if $\\ell(s{w})<\\ell({w})$,}\n \\end{cases}\n\\end{equation}\nfor $s\\in S$, ${w}\\in W$. In particular, one has a canonical isomorphism\n$\\ca{H}_1\\simeq R[W]$, where $R[W]$ denotes the $R$-group algebra of $W$. \nThe $R$-algebra $\\ca{H}$ comes equipped with an antipodal map \n$\\argu^\\natural\\colon\\ca{H}\\longrightarrow\\ca{H}^{\\op}$, $T_{w}^\\natural=T_{{w}^{-1}}$,\ni.e., $\\argu^\\natural$ is an isomorphism satisfying $\\argu^{\\natural\\antip}=\\iid_{\\ca{H}}$\n(cf. \\cite[Chap.~7.3, Ex.~1]{humphreys--rgcg}).\n\nFor $I\\subseteq S$ we denote by $\\ca{H}_I$ the corresponding parabolic \nsubalgebra, i.e., the $\\ca{H}$-subalgebra of $\\ca{H}$ \ngenerated by $\\{\\,T_s\\mid s\\in I\\,\\}$ which coincides with the $R$-module \nspanned by $\\{\\, T_w\\mid w\\in W_I\\,\\}$.\nFor further details see \\cite[Chap.~7]{humphreys--rgcg}.\n\n\\subsection{$\\ca{H}$-modules}\n\\label{ss:one} \nAny $R$-algebra homomorphism $\\lambda\\in\\Hom_{{R\\mbox{-}\\!\\alg}}(\\ca{H},R)$ defines \na $1$-di\\-men\\-sion\\-al left $\\ca{H}$-module $R_\\lambda$, i.e., for $T_{w}\\in\\ca{H}$, \n${w}\\in W$, and $r\\in R_\\lambda$ one has $T_{w}.r=\\lambda(T_{w})r$. \nNote that the relations \\eqref{eq:hecrel} force\n$\\lambda(T_s)\\in \\{-1,q\\}$ \nfor all $s\\in S$.\nMoreover, for $s, s^\\prime\\in S$ and $m(s,s^\\prime)$ odd, one \nhas $\\lambda(T_s)=\\lambda(T_{s^\\prime})$. \nThere are two particular $R$-algebra homomorphisms $\\varepsilon_q$, $\\varepsilon_{-1}\\in\n\\Hom_{{R\\mbox{-}\\!\\alg}}(\\ca{H},R)$, given by $\\varepsilon_q(T_s)=q$, $\\varepsilon_{-1}(T_s)=-1$, $s\\in S$. \nOne may consider $\\varepsilon_q$ as the {\\it augmentation} and $\\varepsilon_{-1}$ as the \n{\\it sign-character}. \nNote that $\\varepsilon_q(T_w)=q^{\\ell(w)}$ and $\\varepsilon_{-1}(T_w)=(-1)^{\\ell(w)}$, and \ntherefore $\\varepsilon_q(T_w)=\\varepsilon_q(T_w^\\natural)$ and \n$\\varepsilon_{-1}(T_w)=\\varepsilon_{-1}(T_w^\\natural)$ for all $w\\in W$.\nFor short we put $R_q=R_{\\varepsilon_q}$, $R_{-1}=R_{\\varepsilon_{-1}}$, and\nuse also the same notation for the restriction of these modules to \nany parabolic subalgebra.\n\nFor $I\\subseteq S$ let $\\ca{H}^I=\\spn_R\\{\\,T_w\\mid w\\in W^I\\,\\}\\subseteq \\ca{H}$.\nMultiplication in $\\ca{H}$ induces a canonical map of right $\\ca{H}_I$-modules\n$\\ca{H}^I\\otimes_R\\ca{H}_I\\longrightarrow\\ca{H}$.\nLet $y\\in W^I$ and $u\\in W_I$. As $\\ell(yu)=\\ell(y)+\\ell(u)$ \n(cf. Prop.~\\ref{prop:cosreps}(e)), one has $T_yT_u= T_{yu}$. \nThis shows that this map is an isomorphism. \nIn particular, $\\ca{H}$ is a projective and thus a flat right $\\ca{H}_I$-module. \nThis implies that\n\\begin{equation}\n \\label{eq:ind}\n \\Mind_I^S(\\argu) =\\Mind_{\\ca{H}_I}^{\\ca{H}}(\\argu) =\n \\ca{H}\\otimes_{\\ca{H}_I}\\argu\\colon{\\caH_I\\mbox{-}\\!\\mmod}\\longrightarrow{\\caH\\mbox{-}\\!\\mmod}\n\\end{equation}\nis an exact functor mapping projectives to projectives. \nMoreover, one has the following.\n\n\\begin{fact}\n \\label{fact:decoH}\n The canonical map $c_I\\colon\\ca{H}^I\\longrightarrow\\Mind_I^S(R_q)$\n given by \n $c_I(T_w)=T_w\\eta_I$, where $\\eta_I=T_1\\otimes 1\\in \\Mind_I^S(R_q)$ \n and $w\\in W^I$, is an isomorphism of $R$-modules.\n Moreover, for $w\\in W$, one has $T_w\\eta_I=\\varepsilon_q(T_{w_I})T_{w^I}\\eta_I$.\n\\end{fact}\n\nIn case that $I\\subseteq S$ generates a finite group, one has\nthe following.\n\n\\begin{prop}\n \\label{prop:eI}\n Let $I$ be a subset of $S$ such that $W_I$ is finite. \n Put $\\tau_I=\\sum_{w\\in W_I}T_w$. Then one has the following:\n \\begin{itemize}\n \\item[(a)] $\\tau_I^2=p_{(W_I,I)}(q)\\tau_I$.\n \\end{itemize}\n Moreover if $p_{(W_I,I)}(q)\\in R^\\times$ is invertible in $R$, let \n $e_I=(p_{(W_I,I)}(q))^{-1}\\tau_I$. Then\n \\begin{itemize}\n \\item[(b)] the element $e_I$ is a central idempotent in $\\ca{H}_I$ \n satisfying $e_I^\\natural=e_I$,\n \\item[(c)] the left ideal $\\ca{H} e_I$ is a finitely generated, \n projective, left $\\ca{H}$-module isomorphic to $\\Mind_I^S(R_q)$,\n \\item[(d)] $T_w e_I=\\varepsilon_q(T_{w_I})T_{w^I}e_I$.\n \\end{itemize}\n\\end{prop}\n\\begin{proof} For $s\\in I$, put $X_s=\\sum_{w\\in {}^{\\{s\\}}(W_I)}T_w$. \n Then $\\tau_I=(T_1+T_s)X_s$ (cf. Prop.~\\ref{prop:cosreps}(f)) \n and therefore \n \\begin{equation*}\n T_s\\tau_I= T_s (T_1 + T_s)X_s=\n \\left[T_s + q T_1 + (q-1)T_s\\right]X_s=q(T_1+T_s)X_s=\\varepsilon_q(T_s) \\tau_I.\n \\end{equation*}\n This shows (a). Part (b) is an immediate consequence of (a), and the first \n part of (c) follows from the decomposition of the regular module \n $\\ca{H} = \\ca{H} e_I \\oplus \\ca{H} (T_1-e_I)$.\n The canonical map $\\pi\\colon\\ca{H}\\longrightarrow \\Mind^S_I(R_q)$, $\\pi(T_w)=T_w\\eta_I$,\n is a surjective morphism of $\\ca{H}$-modules with $\\kernel(\\pi)=\\ca{H}(T_1-e_I)$.\n This yields the second part of (c).\n Part (d) follows from part (b) and Proposition~\\ref{prop:cosreps}(a).\n\\end{proof}\n\n\n\\section{The Deodhar complex}\n\\label{ss:HCcomplex}\nThere is a chain complex of left $\\ca{H}$-modules\n$C=(C_\\bullet,\\partial_\\bullet)$ which can be seen as \nthe module-theoretic analogue of the Coxeter complex\nassociated with a Coxeter group $(W,S)$.\nThis chain complex has been introduced first by\nV.V.~Deodhar in \\cite{deodhar--sgabo2}.\nFor spherical Coxeter groups \nit was studied in more detail by A.~Mathas in \\cite{mathas--qacc}.\nRecently, M.~Linckelmann and S.~Schroll introduced a two-sided version\nof this complex for spherical Coxeter groups (cf. \\cite{linckelmann-schroll--tsqacc}).\nThe definition of this chain complex is quite technical\nand depends on the choice of a sign function,\nand one may speculate that this is the reason why it has not found its way\nin the standard literature yet. In this section \nwe recall its definition and basic properties for the convenience of the reader.\n\n\\subsection{Sign maps}\n\\label{ss:sign}\nLet $\\ca{H}=\\ca{H}_q(W,S)$ be an $R$-Hecke algebra, \nand let $\\ca{P}(S)$ denote the set of subsets of $S$.\nA \\emph{sign map} for $\\ca{H}$ is a function \n$\\sgn\\colon S\\times\\ca{P}(S)\\longrightarrow\\{\\pm1\\}$ satisfying \n\\begin{equation}\\label{eq:signprop}\n\\sgn(s,I)\\sgn(t,I\\sqcup\\{s\\})+\\sgn(t,I)\\sgn(s,I\\sqcup\\{t\\})=0\n\\end{equation}\nfor all $I\\subseteq S$ and $s, t\\in S\\setminus I$, $s\\not=t$.\nE.g., if ``$<$'' is a total order on the finite set $S$, the function \n$\\sgn(s,I)=(-1)^{|\\{\\,t\\in S\\setminus I \\, \\mid\\, t |S|-1$. \nThe following properties have been established in\n\\cite[Thm.~5.1]{deodhar--sgabo2}.\n\n\\begin{thm}[V.V.~Deodhar]\n\\label{thm:complex}\nLet $\\ca{H}=\\ca{H}_q(W,S)$ be an $R$-Hecke algebra, and let \n$C=(C_\\bullet, \\partial_\\bullet)$ be as described in \\textup{\\eqref{eq:derdef2}}. Then\n\\begin{itemize}\n\\item[{\\rm (a)}] $\\partial_k\\circ \\partial_{k+1}=0$, i.e., $C$ is a chain complex.\n\\item[{\\rm (b)}] If $W$ is finite, then \n \\begin{equation*}\n H_k(C)\\simeq \\begin{cases}\n R_q & \\text{ for }k=0\\\\\n R_{-1} & \\text{ for }k=|S|-1\\\\\n 0 & \\text{ otherwise.}\n \\end{cases}\n \\end{equation*}\n\\item[{\\rm (c)}] If $W$ is infinite, then $C$ is acyclic with $H_0(C)\\simeq R_q$. \n\\end{itemize}\n\\end{thm}\n\nFrom now on we will refer to the complex\n$C=(C_\\bullet, \\partial_\\bullet)$ as the {\\it Deodhar complex} of $\\ca{H}$.\n\n\\begin{rem}\n \\label{rem:complex}\n Let $C=(C_\\bullet,\\partial_\\bullet)$ be the Deodhar complex of $\\ca{H}$.\n \n \\noindent\n (a) In degree $|S|-1$, \n $C_{|S|-1}=\\Mind_\\emptyset^S(R_q)\\simeq \\ca{H}$ \n coincides with the regular left $\\ca{H}$-module.\n \n \\noindent\n (b) Let $\\varepsilon\\colon C_0\\longrightarrow R_q=\\Mind_S^S(R_q)$ denote the canonical map\n given by \\eqref{eq:derdef2}. Then the chain complex\n \\begin{equation}\n \\label{eq:trivhom}\n \\xymatrix{\n \\ldots\\ar[r]&C_2\\ar[r]^{\\partial_2}&C_1\\ar[r]^{\\partial_1}&C_0\\ar[r]^{\\varepsilon}&R_q\\ar[r]&0}\n \\end{equation}\n has trivial homology for $W$ infinite.\n\\end{rem}\n\\section{Euler algebras}\n\\label{s:Euler}\nLet $R$ be a commutative ring with unit.\nIn this section we will investigate a class of associative \n$R$-algebras for which one has a natural notion\nof {\\it Euler characteristic}. For this reason we call\nsuch algebras {\\it Euler algebras}.\n\nLet $\\mathbf{A}$ be an associative $R$-algebra\n(with unit $1\\in\\mathbf{A}$).\nAn $R$-linear isomorphism $\\argu^\\natural\\colon \\mathbf{A}\\longrightarrow\\mathbf{A}^{\\op}$ will be \ncalled an {\\it antipode}, if $\\argu^{\\natural\\antip}=\\iid_{\\mathbf{A}}$. \nIf $(\\mathbf{A},\\argu^\\natural)$ is an $R$-algebra with antipode\nthen $\\lambda\\in\\Hom_{{R\\mbox{-}\\!\\alg}}(\\mathbf{A},R)$ will be called an \\emph{augmentation} if\n$\\lambda(a)=\\lambda(a^\\natural)$ for all $a\\in\\mathbf{A}$. Note that $\\lambda$ defines\na left $\\mathbf{A}$-module $R_\\lambda$ which is as set equal to $R$\nand satisfies $a.r=\\lambda(a)r$ for $a\\in\\mathbf{A}$ and $r\\in R_\\lambda$.\n\n\\subsection{Trace functions}\n\\label{ss:trace}\nLet $\\mathbf{A}$ be an associative $R$-algebra. A homomorphism of \n$R$-modules $\\tau\\colon \\mathbf{A}\\longrightarrow R$ satisfying $\\tau(ab)=\\tau(ba)$ for all\n$a,b\\in \\mathbf{A}$ is called a \\emph{trace function} on $\\mathbf{A}$. Let\n$[\\mathbf{A},\\mathbf{A}]=\\spn_R(\\{\\,ab-ba\\mid a,b\\in \\mathbf{A}\\,\\})$,\nand let $\\underline{\\boA}$ denote the $R$-module $\\mathbf{A}\/[\\mathbf{A},\\mathbf{A}]$%\n\\footnote{In the standard literature (cf. \\cite{bass--tec}, \\cite{bass--eccdg},\n \\cite{brown--cg}) this $R$-module is denoted by $T(\\mathbf{A})$.}.\nThen $\\underline{\\boA}^\\ast= \\Hom_R(\\underline{\\boA},R)$ is the $R$-module of all trace \nfunctions of $\\mathbf{A}$.\nThe following elementary property will be useful for our purpose.\n\n\\begin{lem}\n \\label{lem:trace}\n Let $(\\mathbf{A},\\argu^\\natural,\\lambda)$ be an augmented, associative $R$-algebra with antipode, and \n let $\\mathcal{B}\\subset \\mathbf{A}$ be a free generating system of the $R$-module $\\mathbf{A}$ \n with the following properties:\n \\begin{itemize}\n \\item[{\\rm (i)}] $1\\in\\mathcal{B}$;\n \\item[{\\rm (ii)}] $\\mathcal{B}^\\natural=\\mathcal{B}$;\n \\item[{\\rm (iii)}] the symmetric $R$-bilinear form\n \\begin{equation}\n \\label{eq:assform}\n \\langle\\argu,\\argu\\rangle\\colon \\mathbf{A}\\times \\mathbf{A}\\longrightarrow R,\\qquad\n \\langle a,b\\rangle=\\delta_{a,b} \\lambda(a),\\qquad a,b\\in \\mathcal{B},\n \\end{equation}\n where $\\delta_{\\argu,\\argu}$ denotes Kronecker's $\\delta$-function, satisfies\n \\begin{equation}\n \\label{eq:asso}\n \\langle ab,c\\rangle=\\langle b,a^\\natural c\\rangle \\qquad\\text{for all $a,b,c\\in\\mathbf{A}$.}\n \\end{equation}\n \\end{itemize}\n Then $\\tilde\\mu\\in\\Hom_R(\\mathbf{A},R)$ given by\n $\\tilde\\mu(a)=\\langle 1,a\\rangle$, $a\\in\\mathbf{A}$, is a trace function.\n\\end{lem}\n\n\\begin{proof}\n By definition, one has for all $a,b\\in\\mathbf{A}$ that\n $\\langle a^\\natural,b^\\natural\\rangle= \\langle a,b\\rangle$. Hence\n \\begin{equation}\n \\label{eq:trcheck}\n \\tilde\\mu(ab-ba)=\\langle 1,ab\\rangle- \\langle 1,ba\\rangle=\n \\langle a^\\natural,b\\rangle- \\langle b^\\natural,a\\rangle=0.\n \\end{equation}\n for all $a,b\\in \\mathbf{A}$. This yields the claim.\n\\end{proof}\n\n If $(\\mathbf{A},\\argu^\\natural,\\lambda,\\mathcal{B})$ \n satisfies the hypothesis of Lemma~\\ref{lem:trace}, \n the map $\\mu\\in\\Hom_R(\\underline{\\boA},R)$ induced by $\\tilde\\mu$ can be seen as the \n \\emph{canonical trace function} associated with\n $(\\mathbf{A},\\argu^\\natural,\\lambda,\\mathcal{B})$. \n\n\\subsection{Euler algebras}\n\\label{ss:eaec} \nLet $\\mathbf{A}$ be an associative $R$-algebra.\nA left $\\mathbf{A}$-module $M$ is called \\emph{of type {\\rm FP}}, if it has\na finite, projective resolution $(P_\\bullet,\\partial_\\bullet^P,\\varepsilon_M)$,\n$\\varepsilon\\colon P_0\\longrightarrow M$, by finitely generated projective left $\\mathbf{A}$-modules,\ni.e., there exists a positive integer $m$ such that\n$P_k=0$ for $k>m$ or $k<0$, and $P_k$ is finitely generated for all $k$.\nAn augmented, associative $R$-algebra with antipode \n$\\mathbf{A}=(\\mathbf{A},\\argu^\\natural, \\lambda)$ is called of \\emph{type {\\rm FP}}, if\nthe left $\\mathbf{A}$-module $R_\\lambda$, is of type {\\rm FP}. \nWe call an augmented, associative $R$-algebra with antipode\n$(\\mathbf{A},\\argu^\\natural,\\lambda,\\mathcal{B})$ of type {\\rm FP}\\ \nwith a free $R$-basis $\\mathcal{B}$ satisfying the hypothesis of Lemma~\\ref{lem:trace}\nan \\emph{Euler algebra}. \n\n\n\\subsection{Hattori-Stallings trace maps}\n\\label{ss:hatsta}\nFor a finitely generated, projective, left $\\mathbf{A}$-module $P$ let\n$P^\\ast=\\Hom_{\\mathbf{A}}(P,\\mathbf{A})$. Then $P^\\ast$ carries canonically the \nstructure of a right $\\mathbf{A}$-module, and it is also finitely generated \nand projective. One has a canonical isomorphism\n$\\gamma_P\\colon P^\\ast\\otimes_{\\mathbf{A}} P\\longrightarrow \\End_{\\mathbf{A}}(P)$\ngiven by $\\gamma_P(p^\\ast\\otimes p)(q)=p^\\ast(q) p$, $p^\\ast\\in P^\\ast$, $p,q\\in P$\n(cf. \\cite[Chap.~I, Prop.~8.3]{brown--cg}).\nThe {\\it evaluation map} $\\eval_P\\colon P^\\ast\\otimes_{\\mathbf{A}} P\\longrightarrow\\underline{\\boA}$\nis given by $\\eval_P(p^\\ast\\otimes p)=p^\\ast(p)+[\\mathbf{A},\\mathbf{A}]$. The map\n\\begin{equation}\n \\label{eq:hatsta}\n \\trace_P=\\eval_P\\circ\\gamma_P^{-1}\\colon\\End_{\\mathbf{A}}(P)\\longrightarrow\\underline{\\boA}\n\\end{equation}\nis called the \\emph{Hattori--Stallings trace map on $P$}\nand $r_P=\\trace_P(\\iid_P)\\in\\underline{\\boA}$ the\n\\emph{Hattori--Stallings rank} of $P$ (cf. \\cite{stallings--cg}, \n\\cite[Chap.~IX.2]{brown--cg}). \nIn particular, $\\trace_P$ is $R$-linear, and for $f,g\\in\\End_{\\mathbf{A}}(P)$ one has\n\\begin{equation}\n \\label{eq:hatstatr}\n \\trace_P(f\\circ g)=\\trace_P(g\\circ f).\n\\end{equation}\nFrom the elementary properties of the evaluation map\none concludes that if $P_1$ and $P_2$ are two finitely generated \nprojective left $\\mathbf{A}$-modules, one has\n\\begin{equation}\n \\label{eq:addr}\n r_{P_1\\oplus P_2}=r_{P_1}+r_{P_2}.\n\\end{equation}\nLet $e \\in \\mathbf{A}$, $e=e^2$, be an idempotent in the $R$-algebra $\\mathbf{A}$. \nThen $\\mathbf{A} e$ is a finitely generated, projective, left \n$\\mathbf{A}$-module, and \n\\begin{equation}\\label{eq:r-idemp}\n r_{\\mathbf{A} e}=e+[\\mathbf{A},\\mathbf{A}].\n\\end{equation}\n\n\\subsection{Finite, projective chain complexes}\n\\label{ss:ccs}\nA chain complex $P=(P_\\bullet, \\partial^P_\\bullet)$ of left $\\mathbf{A}$-modules will be\ncalled \\emph{finite} if $\\{\\, k\\in\\mathbb{Z}\\mid P_k\\not=0\\,\\}$ is finite\nand $P_k$ is finitely generated for all $k\\in \\mathbb{Z}$. Moreover, $P$ will be called \n\\emph{projective}, if $P_k$ is projective for all $k$. \n\nFor $P=(P_\\bullet, \\partial^P_\\bullet)$ and $Q=(Q_\\bullet, \\partial^Q_\\bullet)$ finite,\nprojective chain complexes of left $\\mathbf{A}$-modules we denote by \n$(\\dHom_{\\mathbf{A}}(P,Q)_\\bullet,d_\\bullet)$ the chain complex of \nright $\\mathbf{A}$-modules\n\\begin{equation}\n \\label{eq:Chomcom}\n \\dHom_{\\mathbf{A}}(P,Q)_k=\n \\prod_{j=i+k}\\Hom_{\\mathbf{A}}(P_i,Q_j),\n\\end{equation}\nwith differential given by\n\\begin{equation}\n \\label{eq:Chomcom2}\n (d_k(f_k))_{i,j-1}=\n \\partial^Q_{j}\\circ f_{i,j}-(-1)^k f_{i-1,j-1}\\circ \\partial_i^P,\n\\end{equation}\nfor $f_k=\\sum_{j=i+k}f_{i,j}$.\nIn particular, $f_0=\\sum_{i\\in\\mathbb{Z}} f_{i,i}\\in\\dHom_{\\mathbf{A}}(P,Q)_0$ is a\nchain map of degree $0$ if, and only if, $f_0\\in\\kernel(d_0)$, \nand $f_0$ is homotopy equivalent to the $0$-map if, and only if,\n$f_0\\in\\image(d_1)$ (cf. \\cite[Chap.~I]{brown--cg}).\nPut $\\dExt^{\\mathbf{A}}_0(P,Q)=H_{0}(\\dHom_{\\mathbf{A}}(P,Q))$.\n\nLet $B=(B_\\bullet,\\partial_\\bullet^B)$ be a finite, projective chain complex \nof right $\\mathbf{A}$-modules. Then $(B\\utimes_{\\mathbf{A}} P,\\partial_\\bullet^{\\utimes})$\ndenotes the complex\n\\begin{equation}\n \\label{eq:Cdeften}\n \\begin{gathered}\n (B\\utimes_{\\mathbf{A}} P)_k=\\coprod_{i+j=k} B_i\\otimes_{\\mathbf{A}}\n P_j,\\\\\n \\partial_{i+j}^{\\utimes}(b_i\\otimes p_j)=\\partial_i^B(b_i)\\otimes p_j\n +(-1)^i b_i\\otimes \\partial^P_j(p_j).\n \\end{gathered}\n\\end{equation}\nLet $\\mathbf{A}\\llbracket 0\\rrbracket$ denote the chain complex of left $\\mathbf{A}$-modules \nconcentrated in degree $0$ with $\\mathbf{A}\\llbracket 0\\rrbracket_0=\\mathbf{A}$, and let \n$\\underline{\\boA}\\llbracket 0\\rrbracket$ denote the chain complex of $R$-modules\nconcentrated in degree $0$ with $\\underline{\\boA}\\llbracket 0\\rrbracket_0=\\underline{\\boA}$.\nThen $P^\\circledast=(P^\\circledast_\\bullet,\\partial_\\bullet^{P^\\circledast})=\n(\\dHom_\\mathbf{A}(P,\\mathbf{A}\\llbracket 0\\rrbracket)_\\bullet,d_\\bullet)$,\n\\begin{equation}\n \\label{eq:Cdefhomcom}\n \\begin{gathered}\n P^\\circledast_{k}=\\Hom_\\mathbf{A}(P_{-k},\\mathbf{A}),\\\\\n \\partial_k^{P^\\circledast}(p_k^\\ast)(p_{1-k})=(-1)^{k+1} p^\\ast_k(\\partial^P_{1-k}(p_{1-k})).\n \\end{gathered}\n\\end{equation}\nis a finite, projective complex of right $\\mathbf{A}$-modules.\nNote that the differential of the complex is chosen in such a way that the\n\\emph{standard evaluation mapping}\n\\begin{equation}\n \\label{eq:Csteva}\n \\begin{gathered}\n \\eval_P\\colon\n P^\\circledast \\utimes_{\\mathbf{A}} P\\longrightarrow \\underline{\\boA}\\llbracket 0\\rrbracket,\\\\\n \\eval_{s,t}(p^\\ast_s\\otimes p_t)=\\delta_{s+t,0}\\,\\,p^\\ast_s(p_t) ,\n \\end{gathered}\n\\end{equation}\nis a mapping of chain complexes.\nHowever, the natural isomorphism\n\\begin{equation}\n \\label{eq:Chomcomten}\n \\begin{gathered}\n \\gamma\\colon\n \\dHom_\\mathbf{A}(\\argu_1,\\mathbf{A}\\llbracket 0\\rrbracket)\\utimes_\\mathbf{A}\\argu_2\\longrightarrow\n \\dHom_\\mathbf{A}(\\argu_1,\n \\argu_2)\\\\\n \\gamma_{s,t}(p_s^\\ast\\otimes_\\mathbf{A} q_t)(x_{-s})=(-1)^{st}\\,\n p_s^\\ast(x_{-s}) q_t\n \\end{gathered}\n\\end{equation}\ncomes equipped with a non-trivial sign\n(cf. \\cite[Chap.~I, Prop.~8.3(b) and Chap.~VI, \\S6, Ex.~1]{brown--cg}).\nIn this context the \\emph{Hattori--Stallings trace map} is given by\n\\begin{equation}\n \\label{eq:tracecom}\n \\trace_P=H_0(\\eval_P\\circ\\gamma^{-1}_{P,P})\\colon\\dExt_0^{\\mathbf{A}}(P,P)\\longrightarrow\\underline{\\boA}.\n\\end{equation}\nIt has the following properties:\n\n\\begin{prop}\n \\label{prop:tracecc}\n Let $P=(P_\\bullet,\\partial_\\bullet^P)$ be a finite, projective complex of left \n $\\mathbf{A}$-modules, and let $[f],[g]\\in\\dExt_0^{\\mathbf{A}}(P,P)$, $f=\\sum_{k\\in\\mathbb{Z}}f_k$,\n be homotopy classes of chain maps of degree $0$. Then\n \\begin{itemize}\n \\item[(a)] $\\trace_P([f])=\\sum_{k\\in\\mathbb{Z}}(-1)^k \\trace_{P_k}(f_k)$;\n \\item[(b)] $\\trace_P([f]\\circ [g])=\\trace_P([g]\\circ [f])$.\n \\item[(c)] Let $Q=(Q_\\bullet,\\partial_\\bullet^Q)$ be another finite, projective \n complex of left $\\mathbf{A}$-modules which is homotopy equivalent to $P$, i.e.,\n there exist chain maps $\\phi\\colon P\\longrightarrow Q$, $\\psi\\colon Q\\longrightarrow P$, which \n composites are homotopy equivalent to the respective identity maps. \n Let $[h]\\in\\dExt_0^{\\mathbf{A}}(Q,Q)$ such that $[\\phi]\\circ [f]=[h]\\circ[\\phi]$. \n Then $\\trace_P([f])=\\trace_Q([h])$.\n \\end{itemize}\n\\end{prop}\n\n\\begin{proof}\n Part (a) is a direct consequence of \\eqref{eq:Chomcomten}, \n and (b) follows from (a) and \\eqref{eq:hatstatr}.\n\n \\noindent\n The left hand side quadrangle in the diagram\n \\begin{equation}\n \\label{dia:mass}\n \\xymatrix{\n \\dHom_{\\mathbf{A}}(P,P) \\ar[d]_{\\phi\\circ\\argu\\circ\\psi}& \\ar[l]_-{\\gamma}\n P^\\circledast\\utimes_{\\mathbf{A}} P\\ar[r]^-{\\eval_P}\\ar[d]^{\\psi^\\circledast\\otimes\\phi}&\n \\underline{\\boA}\\llbracket 0\\rrbracket\\ar@{=}[d]\\\\\n \\dHom_{\\mathbf{A}}(Q,Q)& \\ar[l]_-{\\gamma} \n Q^\\circledast\\utimes_{\\mathbf{A}} Q\\ar[r]^-{\\eval_Q}&\n \\underline{\\boA}\\llbracket 0\\rrbracket\\\\\n }\n \\end{equation}\n commutes, and the right hand side quadrangle commutes\n up to homotopy equivalence. This yields claim~(c).\n\\end{proof}\n\nLet $P=(P_\\bullet,\\partial_\\bullet^P)$ be a finite, projective complex \nof left $\\mathbf{A}$-modules. Then one defines \nthe \\emph{Hattori--Stallings rank} of $P$ by\n\\begin{equation}\n \\label{eq:hatstacom}\n r_P=\\trace_P([\\iid_P])=\\textstyle{\\sum_{k\\in\\mathbb{Z}} (-1)^k r_{P_k}.}\\in\\underline{\\boA}\n\\end{equation}\nProposition~\\ref{prop:tracecc} implies that if $Q=(Q_\\bullet,\\partial_\\bullet^Q)$\nis another finite, projective, complex of left $\\mathbf{A}$-modules which is \nhomotopy equivalent to $P$ then $r_P=r_Q$.\n\nLet $\\ca{K}(\\mathbf{A})$ denote the additive category the objects of which are\nfinite, projective chain complexes of left $\\mathbf{A}$-modules.\nMorphisms $\\Hom_{\\ca{K}(\\mathbf{A})}(P,Q)= \\dExt_0^\\mathbf{A}(P,Q)$\nare given by the homotopy classes of chain maps of degree $0$.\nIn particular, $\\ca{K}(\\mathbf{A})$ is a \\emph{triangulated category} and distinguished \ntriangles are triangles isomorphic to the cylinder\/cone triangles \n(cf. \\cite{gelfand-manin--ha}, \\cite[Chap.~10]{weibel--iha}). \nThus, if\n\\begin{equation}\n \\label{eq:distri}\n \\xymatrix{\n A\\ar[r]& B\\ar[r]& C\\ar[r]& A[1]\n }\n\\end{equation}\nis a distinguished triangle in $\\ca{K}(\\mathbf{A})$, one has $r_B=r_A+r_C$.\n\nLet $M$ be a left $\\mathbf{A}$-module of type {\\rm FP}, and let\n$(P_\\bullet,\\partial_\\bullet,\\varepsilon_M)$ be a finite, projective resolution.\nIn particular, $P=(P_\\bullet,\\partial_\\bullet)$ is a finite, projective chain complex\nof left $\\mathbf{A}$-modules.\nOne defines the Hattori--Stallings rank of $M$ by\n$r_M=r_P\\in\\underline{\\boA}$.\nThe comparison theorem in homological algebra implies that this element \nis well defined.\nThe following property will be useful for our purpose.\n\n\\begin{prop}\n \\label{prop:FPcc}\n Let $C=(C_\\bullet,\\partial^C_\\bullet)$ be a chain complex of left $\\mathbf{A}$-modules concentrated\n in non-negative degrees with the following\n properties:\n \\begin{itemize}\n \\item[(a)] $C$ is acyclic, i.e., $H_k(C)=0$ for $k\\in\\mathbb{Z}$, $k\\not=0$;\n \\item[(b)] $C$ is finitely supported, i.e., $C_k=0$ for almost all $k\\in\\mathbb{Z}$;\n \\item[(c)] $C_k$ is of type {\\rm FP}\\ for all $k\\in\\mathbb{Z}$.\n \\end{itemize}\n Then $H_0(C)$ is of type {\\rm FP}, and one has\n \\begin{equation}\n \\label{eq:idCs}\n r_{H_0(C)}=\\textstyle{\\sum_{k\\geq 0} (-1)^k r_{C_k}\\in\\underline{\\boA}.}\n \\end{equation}\n\\end{prop}\n\n\\begin{proof}\n Let $\\ell(C)=\\min\\{\\,n\\geq 0\\mid C_{n+j}=0\\ \\text{for all $j\\geq 0$}\\,\\}$ \n denote the length of $C$. We proceed by induction on $\\ell(C)$. \n For $\\ell(C)=1$, there is nothing to prove.\n Suppose the claim holds for chain complexes $D$, $\\ell(D)\\leq \\ell-1$, \n satisfying the hypothesis (a)--(c), and let $C$ be a complex satisfying \n (a)--(c) with $\\ell(C)=\\ell$.\n Let $C^\\wedge$ be the chain complex coinciding with $C$ in all \n degrees $k\\in\\mathbb{Z}\\setminus\\{0\\}$ and satisfying $C^\\wedge_0=0$. \n Then $C^\\wedge[-1]$ satisfies (a)--(c) and $\\ell(C^\\wedge[-1])\\leq \\ell-1$.\n Then by induction, $M=H_1(C^\\wedge)=H_0(C^\\wedge [-1])$ is of type {\\rm FP},\n and $r_M=\\sum_{k\\geq 1} (-1)^{k+1} r_{C_k}$. \n By construction, one has a short exact sequence\n of left $\\mathbf{A}$-modules $0\\longrightarrow M\\overset{\\alpha}{\\longrightarrow} C_0\\longrightarrow H_0(C)\\longrightarrow 0$.\n Let $(P_\\bullet,\\partial_\\bullet^P,\\varepsilon_M)$ be a finite, projective resolution of $M$,\n and let $(Q_\\bullet,\\partial_\\bullet^Q,\\varepsilon_{C_0})$ be a finite, projective \n resolution of $C_0$. By the comparison theorem in homological algebra, \n there exists a chain map $\\alpha_\\bullet\\colon P_\\bullet \\longrightarrow Q_\\bullet$\n inducing $\\alpha$. \n Let $\\Cone(\\alpha_\\bullet)$ denote the mapping cone of $\\alpha_\\bullet$.\n Then $(\\Cone(\\alpha_\\bullet),\\tilde{\\partial}_\\bullet,\\varepsilon_\\ast)$ is a finite, \n projective resolution of $H_0(C)$, i.e., $H_0(C)$ is of type {\\rm FP}.\n Moreover, by the remark following \\eqref{eq:distri} one has\n \\begin{equation}\n \\label{eq:indsum}\n r_{H_0(C)}=r_{\\Cone(\\alpha_\\bullet)}=r_Q-r_P=r_{C_0}-r_M.\n \\end{equation}\n This yields the claim.\n\\end{proof}\n\n\\subsection{The Euler characteristic of an Euler algebra}\n\\label{ss:eueuc}\nLet $\\mathbf{A}=(\\mathbf{A},\\argu^\\natural,\\lambda,\\mathcal{B})$ be an Euler $R$-algebra\nwith canonical trace function $\\mu\\in\\Hom_R(\\underline{\\boA},R)$. We define\nthe \\emph{Euler characteristic} of $\\mathbf{A}$ by\n\\begin{equation}\n \\label{eq:eucharA}\n \\chi_{\\mathbf{A}}=\\chi_{(\\mathbf{A},\\argu^\\natural,\\lambda,\\mathcal{B})}=\\mu(r_{R_\\lambda})\\in R.\n\\end{equation}\n\n\n\\subsection{Induction}\n\\label{ss:ind}\nLet $\\mathbf{B}\\subseteq \\mathbf{A}$ be an $R$-subalgebra of $\\mathbf{A}$.\nThe canonical injection $j\\colon \\mathbf{B}\\longrightarrow\\mathbf{A}$ induces a canonical map\n\\begin{equation}\n \\label{eq:traceBA}\n \\trace_{\\mathbf{B}\/\\mathbf{A}}\\colon\\underline{\\boB}\\longrightarrow\\underline{\\boA}.\n\\end{equation}\nInduction $\\Mind_{\\mathbf{B}}^{\\mathbf{A}} =\\mathbf{A}\\otimes_{\\mathbf{B}}\\argu$\nis a covariant additive right-exact functor mapping finitely generated\nprojective left $\\mathbf{B}$-modules to finitely generated projective left \n$\\mathbf{A}$-modules.\nMoreover, if $\\mathbf{A}$ is a flat right $\\mathbf{B}$-module, then \n$\\Mind_{\\mathbf{B}}^{\\mathbf{A}}$ is exact. \nLet $P$ be a finitely generated left $\\mathbf{B}$-module,\nand let $Q=\\Mind_{\\mathbf{B}}^{\\mathbf{A}}(P)$. Then one has a canonical map \n$\\iota\\colon P\\longrightarrow Q$, $\\iota(p)=1\\otimes p$, which is a homomorphism \nof left $\\mathbf{B}$-modules.\nAs induction is left adjoint to restriction, every map $f\\in\\End_{\\mathbf{B}}(P)$ \ninduces a map $\\iota_\\circ(f)=(\\iota\\circ f)_\\ast\\in\\End_{\\mathbf{A}}(Q)$.\n\nLet $P^\\ast=\\Hom_\\mathbf{B}(P,\\mathbf{B})$ and $Q^\\ast=\\Hom_{\\mathbf{A}}(Q,\\mathbf{A})$.\nThen for $f\\in\\Hom_{\\mathbf{B}}(P,\\mathbf{B})$ one has an induced map\n$\\iota_\\ast(f)=(j\\circ f)_\\ast\\in Q^\\ast$\nmaking the\ndiagram\n\\begin{equation}\n \\xymatrix{\n\\End_{\\mathbf{B}}(P)\\ar[d]_{\\iota_\\circ}&P^\\ast\\otimes_{\\mathbf{B}} P\n\\ar[l]_-{\\gamma_P}\\ar[d]^{\\iota_\\ast\\otimes\\iota}\\ar[r]^-{\\eval_P} &\\underline{\\boB}\\ar[d]^{\\trace_{\\mathbf{B}\/\\mathbf{A}}}\\\\\n\\End_{\\mathbf{A}}(Q)&Q^\\ast\\otimes_{\\mathbf{A}}Q\n\\ar[l]_-{\\gamma_Q}\\ar[r]^-{\\eval_Q}&\\underline{\\boA}\n }\n\\end{equation}\ncommute. This shows the following.\n\n\\begin{prop}\n \\label{prop:ind}\n Let $\\mathbf{B}\\subseteq\\mathbf{A}$ be an $R$-subalgebra of $\\mathbf{A}$ such that\n $\\mathbf{A}$ is a flat right $\\mathbf{B}$-module, and let $M$ be a left $\\mathbf{B}$-module of type {\\rm FP}.\n Then $\\Mind_{\\mathbf{B}}^{\\mathbf{A}}(M)$ is of type {\\rm FP}, and one has\n \\begin{equation}\n \\label{eq:Mindr}\n r_{\\Mind_{\\mathbf{A}}^{\\mathbf{B}}(M)}=\\trace_{\\mathbf{B}\/\\mathbf{A}}(r_M).\n \\end{equation}\n\\end{prop}\n\nLet $(\\mathbf{A}, \\argu^\\natural,\\lambda, \\mathcal{B})$ be an augmented, associative, \n$R$-algebra with antipode and a distinguished $R$-basis $\\mathcal{B}$ satisfying \nthe hypothesis of Lemma~\\ref{lem:trace}. Let $\\mathbf{B}\\subseteq\\mathbf{A}$\nbe an $R$-subalgebra of $\\mathbf{A}$ such that\n\\begin{itemize}\n\\item[(i)] $\\mathbf{A}$ is a flat right $\\mathbf{B}$-module;\n\\item[(ii)] $\\mathbf{B}^\\natural=\\mathbf{B}$;\n\\item[(iii)] $\\ca{C}=\\mathcal{B}\\cap\\mathbf{B}$ is an $R$-basis of $\\mathbf{B}$.\n\\end{itemize}\nThen the augmented, associative $R$-algebra\n$(\\mathbf{B},\\argu^\\natural,\\lambda\\vert_\\mathbf{B},\\ca{C})$ satisfies the hypothesis\nof Lemma~\\ref{lem:trace}.\nLet $\\mu_\\mathbf{A}\\colon\\underline{\\boA}\\longrightarrow R$ and $\\mu_\\mathbf{B}\\colon\\underline{\\boB}\\longrightarrow R$ denote the\nassociated canonical traces. Then one has a commutative diagram\n\\begin{equation}\n \\label{eq:diatritr}\n \\xymatrix{\n \\underline{\\boB}\\ar[0,2]^{\\trace_{\\mathbf{B}\/\\mathbf{A}}}\\ar[dr]_{\\mu_\\mathbf{B}}&&\\underline{\\boA}\\ar[dl]^{\\mu_{\\mathbf{A}}}\\\\\n &R&\n }\n\\end{equation}\nFrom this one concludes the following direct consequence of Proposition~\\ref{prop:ind}.\n\n\\begin{cor}\n \\label{cor:ind}\n Let $(\\mathbf{A}, \\argu^\\natural,\\lambda, \\mathcal{B})$ be an augmented, associative, \n $R$-algebra with antipode and a distinguished $R$-basis $\\mathcal{B}$ satisfying \n the hypothesis of Lemma~\\ref{lem:trace}, and let $\\mathbf{B}\\subseteq\\mathbf{A}$ be an $R$-subalgebra satisfying {\\rm (i)-(iii)}. Let\n $M$ be a left $\\mathbf{B}$-module of type FP. Then\n $\\mu_\\mathbf{B}(r_M)=\\mu_\\mathbf{A}(r_{\\Mind_\\mathbf{B}^\\mathbf{A}(M)})$.\n\\end{cor}\n\\section{The Euler characteristic of a Hecke algebra}\n\\label{s:trhec}\n\n\\subsection{The canonical trace of a Hecke algebra}\n\\label{ss:cantr}\nLet $\\ca{H}=\\ca{H}_q(W,S)$ be the $R$-Hecke algebra associated to the finitely generated\nCoxeter group $(W,S)$, and let $\\mathcal{B}=\\{\\,T_{w}\\mid{w}\\in W\\,\\}$.\nThen $\\argu^\\natural\\colon\\ca{H}\\longrightarrow\\ca{H}^{\\op}$, $T_{w}^\\natural=T_{{w}^{-1}}$,\nis an anti-automorphism of $\\ca{H}$ satisfying $\\argu^{\\natural\\antip}=\\iid_{\\ca{H}}$\n(cf. \\cite[Chap.~7.3, Ex.~1]{humphreys--rgcg}) and $\\varepsilon_q(a^\\natural)=\\varepsilon_q(a)$ for all $a\\in\\ca{H}$.\nOne has the following property.\n\n\\begin{prop}\n \\label{prop:mar}\n Let $\\ca{H}$ be the Hecke algebra associated to the finitely generated \n Coxeter group $(W,S)$.\n Then the $R$-bilinear map $\\langle\\argu,\\argu\\rangle\\colon\\ca{H}\\times\\ca{H}\\longrightarrow R$ \n associated to $(\\ca{H},\\argu^\\natural,\\varepsilon_q,,\\mathcal{B})$ satisfies \\eqref{eq:asso}. \n In particular, $\\tilde\\mu_\\mathcal{B}=\\langle\\, T_1,\\argu\\,\\rangle$ is a trace function.\n\\end{prop}\n\n\\begin{proof}\n By Lemma~\\ref{lem:trace}, one has to show that\n \\begin{equation}\n \\label{eq:traceid}\n \\langle\\, T_uT_v,T_{w}\\,\\rangle=\\langle\\, T_v,T_{u^{-1}}T_{w}\\,\\rangle\\qquad \n \\text{for all $u,v,{w}\\in W$.}\n \\end{equation}\n Using induction one easily concludes that it suffices to show \\eqref{eq:traceid}\n in the case that $u=s\\in S$. In this case one has:\n \\begin{align}\n \\lambda=\\langle\\, T_sT_v,T_{w}\\,\\rangle =\\begin{cases}\n \\delta_{sv,{w}}\\varepsilon_q(T_{sv})&\\qquad \\text{if $\\ell(sv)>\\ell(v)$}\\\\\n (q-1)\\delta_{v,{w}}\\varepsilon_q(T_v)+q\\delta_{sv,{w}} \\varepsilon_q(T_{sv}) &\\qquad \\text{if $\\ell(sv)<\\ell(v)$}\n \\end{cases}\\label{eq:lhs}\\\\\n \\intertext{and} \n \\rho=\\langle\\, T_v,T_sT_{w}\\,\\rangle =\\begin{cases}\n \\delta_{v,s{w}}\\varepsilon_q(T_v)&\\qquad \\text{if $\\ell(s{w})>\\ell({w})$}\\\\\n (q-1)\\delta_{v,{w}}\\varepsilon_q(T_v)+q\\delta_{v,s{w}} \\varepsilon_q(T_v) &\\qquad \n \\text{if $\\ell(s{w})<\\ell({w})$}\n \\end{cases}\\label{eq:rhs}\n \\end{align}\n We proceed by a case-by-case analysis. \n\n \\textbf{Case 1:}\n $\\ell(sv)>\\ell(v)$ and $\\ell(s{w})>\\ell({w})$.\n Suppose that $\\lambda\\neq 0$. Then $sv={w}$, but \n $ \\ell({w})=\\ell(sv)>\\ell(v)=\\ell(sw)$, a contradiction. Hence $\\lambda= 0$. \n Reversing the r\\^oles of $v$ and ${w}$ yields $\\lambda=\\rho=0$ and thus the claim.\n\n \\textbf{Case 2:} \n $\\ell(sv)>\\ell(v)$ and $\\ell(s{w})<\\ell({w})$. \n Then, $v\\neq {w}$. \n If $\\lambda \\neq 0$, then $sv={w}$. Hence $\\ell({w})=\\ell(sv)=\\ell(v)+1$, and \n $\\lambda=\\varepsilon_q(T_{w})=\\varepsilon_q(T_s)\\varepsilon_q(T_v)$. On the other hand, \n $\\rho=(q-1)\\delta_{v,{w}}\\varepsilon_q(T_v)+q\\delta_{v,s{w}} \\varepsilon_q(T_v)=\n q\\varepsilon_q(T_v)=\\lambda$.\n If $\\lambda = 0$, then $sv\\neq{w}$. Hence\n $\\rho=(q-1)\\delta_{v,{w}}\\varepsilon_q(T_v)+q\\delta_{v,s{w}} \\varepsilon_q(T_v)=\n 0=\\lambda$.\n\n \\textbf{Case 3:} \n $\\ell(sv)<\\ell(v)$ and $\\ell(s{w})>\\ell({w})$. Reversing the r\\^oles of $v$ and \n ${w}$ one can transfer the proof for Case 2 verbatim.\n\n \\textbf{Case 4:} $\\ell(sv)<\\ell(v)$ and $\\ell(s{w})<\\ell({w})$.\n Suppose that $sv={w}$, or, equivalently, $v=s{w}$. \n Then $\\ell(sv)< \\ell(v)=\\ell(s{w})< \\ell({w})$, a contradiction. \n Hence $sv\\neq{w}$ and $v\\neq s{w}$. Thus\n $\\lambda=\\rho$.\n This completes the proof.\n\\end{proof}\n\n\\begin{rem}\n \\label{rem:trace}\n The trace function $\\tilde\\mu\\colon\\ca{H}\\longrightarrow R$ can be seen as the \n {\\it canonical trace function} on $\\ca{H}$. It is straight forward to \n verify that for Hecke algebras of type $A_n$, $B_n$ or $D_n$, this trace \n function coincides with the Jones--Ocneanu trace evaluated in $0$ \n (cf. \\cite{geck:trace}).\n\\end{rem}\n\n\\subsection{Properties of the Deodhar complex}\n\\label{ss:coxcomp}\nLet $(W,S)$ be a finite Coxeter group, and let $q\\in R$ be \nsuch that $p_{(W,S)}(q)\\in R^\\times$. Then $R_q\\simeq \\ca{H} e_S$ (cf. Prop.~\\ref{prop:eI});\nin particular, $R_q$ is a projective left $\\ca{H}$-module.\nThis shows that for any Coxeter group $(W,S)$ and $I\\subseteq S$, $W_I$ is finite,\n$\\Mind_{\\ca{H}_I}^\\ca{H}(R_q)$ is a finitely generated, projective, left $\\ca{H}$-module.\nAs a consequence one has the following (cf. \\cite[\\S 6.8]{humphreys--rgcg}):\n\n\\begin{prop}\n \\label{prop:latt}\n Let $(W,S)$ be a finitely generated Coxeter group, which is either affine,\n or co-compact hyperbolic (cf. \\cite[Ch. 6]{humphreys--rgcg}), and let $q\\in R$ be such that\n $p_{(W_I,I)}(q)\\in R^\\times$ for any proper parabolic subgroup $(W_I,I)$.\n Then the Deodhar complex $(C_\\bullet,\\partial_\\bullet,\\varepsilon)$ \n together with the map $\\varepsilon\\colon C_0\\longrightarrow R_q$ (cf. Rem.~\\ref{rem:complex}), is a \n finite, projective resolution of $R_q$.\n\\end{prop}\n\n\n\n\\subsection{The Euler characteristic of a Hecke algebra}\n\\label{ss:euhec}\n\n\\begin{proof}[Proof of Theorem A]\nAs $p_{(W_I,I)}(q)\\in R^\\times$ for any finite parabolic subgroup $(W_I,I)$,\n$\\Mind_{\\ca{H}_I}^\\ca{H}(R_q)$ is a finitely generated projective $\\ca{H}$-module\nfor any finite parabolic subgroup $(W_I,I)$.\nFirst we show that $\\ca{H}_q$ is an $R$-Euler algebra.\n We proceed by induction on $d=|S|$.\n For $|S|\\leq 2$, $(W,S)$ is spherical or affine.\n Hence there is nothing to prove (cf. Prop.~\\ref{prop:latt}). Assume that the\n claim holds for all Coxeter groups $(W_J,J)$ with $|J|*\\txt{\\Large \\textbf{A}}\\restore \\ar[r]\\ar[dddd]&j'_!j'^!\\mathbb{Q}_{Y'} \\save[]+<3.7cm,-1.5cm>*\\txt<20pt>{\\Large \\textbf{C}}\n\\restore \\ar[rr]\\ar[d]\\ar[ddr]&&\\mathbb{Q}_{Y'}\\ar[dddd]\n\\\\\n&j'_!g_!g^*j'^!\\mathbb{Q}_{Y'}\\ar[dd]^{j'_!g_!\\upsilon}\\ar[rd]_-{h_!\\varepsilon j'^!}\\save[]+<.5cm,.5cm>*\\txt{\\Large \\textbf{E}}\\restore\n\\\\\n&&h_!h^*j'_!j'^!\\mathbb{Q}_{Y'}\\ar[rdd]\n\\\\\n&j'_!g_!j^!h^*\\mathbb{Q}_{Y'}\\ar[d]^{=}\n\\\\\nj'_!g_!g^*j'^*\\mathbb{Q}_{Y'}[d]\\ar[r]\\ar[uuur]\\save[]+<1.6cm,1.1cm>*\\txt{\\Large \\textbf{B}}\\restore \n&h_!j_!j^!h^*\\mathbb{Q}_{Y'} \\save[]+<2.3cm,2cm>*\\txt{\\Large \\textbf{D}}\\restore \\ar[rr]&&h_!h^*\\mathbb{Q}_{Y'}\n}\n\\]\nThe map $\\varepsilon$ is the base change isomorphism. Applying $(Y'^{\\Sp}\\rightarrow Y')^*\\phi_f$ to the square formed by the corners of this diagram, and taking dual compactly supported cohomology, we obtain diagram (\\ref{descom}). So it will be enough to prove that the labelled sub-diagrams commute.\n\\begin{itemize}\n\\item \\textbf{A} commutes since it is obtained by applying the adjunction $\\id\\rightarrow g_!g^*$ to the morphism $j'^*\\mathbb{Q}_{Y'}[d]\\rightarrow j'^!\\mathbb{Q}_{Y'}$ and applying $j'_!$ to the resulting commutative square.\n\\item Commutativity of \\textbf{B} and \\textbf{D} is given by applying $h_!$ to the commutativity conditions on $\\upsilon$.\n\\item \\textbf{C} commutes since it is obtained by applying the adjunction $\\id\\rightarrow h_!h^*$ to the morphism $j'_!j'^!\\mathbb{Q}_{Y'}\\rightarrow\\mathbb{Q}_{Y'}$.\n\\item The commutativity of \\textbf{E} is an exercise in category theory. We may rewrite the base change map $\\overline{\\varepsilon}^{-1}: h^*j'_*\\rightarrow j_*g^*$ as the composition\n\\[\nh^*j'_*\\rightarrow h^*j'_*g_*g^*=h^*h_*j_*g^*\\rightarrow j_*g^*.\n\\]\nWe claim that the following diagram of natural transformations commutes\n\\[\n\\xymatrix{\n&j'_*\\ar[ddl]\\ar[dr]\n\\\\\n&&h_*h^*j'_*\\ar[d]\n\\\\\nj'_*g_*g^*\\ar[ddr]^=\\ar[rr]&&h_*h^*j'_*g_*g^*\\ar[d]^=\n\\\\\n&&h_*h^*h_*j_*g^*\\ar[dl]\n\\\\\n&h_*j_*g^*.\n}\n\\]\nThe top half commutes since it is obtained by applying the adjunction $\\id\\rightarrow h_*h^*$ to the adjunction $j'_*\\rightarrow j'_*g_*g^*$ and the bottom half commutes because the longer route is obtained by postcomposing the composition $h_*\\rightarrow h_*h^*h_*\\rightarrow h_*=\\id$ with $j_*g^*$. By restriction, we obtain that the diagram of natural transformations \n\\[\n\\xymatrix{\nj'_!\\ar[d]\\ar[dr]\\\\\nj'_!g_!g^*&h_!h^*j'_!\\ar[l]\n}\n\\]\ncommutes, and commutativity of \\textbf{E} follows.\n\n\\end{itemize}\n\n\\end{proof}\n\\begin{corollary}\n\\label{ppCor}\nAssume that we have a Cartesian diagram as in (\\ref{cornersquare}), and that either\n\\begin{enumerate}\n\\item\n$j$ and $j'$ are \\'etale locally trivial fibrations with smooth fibres or\n\\item\n$h$ and $j'$ are inclusions of transversally intersecting submanifolds of $Y'$.\n\\end{enumerate}\nThen the diagram (\\ref{descom}) commutes.\n\\end{corollary}\n\\begin{proof}\nIt is easy to check that we have the required isomorphism $\\upsilon$ and the result follows.\n\\end{proof}\n\\begin{proposition}\n\\label{maneq}\nLet $g:X\\rightarrow Y$ be a $G$-equivariant map of manifolds. Then\n\\begin{itemize}\n\\item\nIf $g$ is a closed embedding, the map $g^*g_*:\\mathrm{H}_{c,G}(X,\\phi_{fg})^*\\rightarrow\\mathrm{H}_{c,G}(X,\\phi_{fg})^*$ is given by multiplication by the Euler class of the normal bundle $\\No_{X\/Y}$\n\\item\nIf $g$ is an affine fibration, and the Euler class $\\eu(g)$ is a non zero divisor in $\\mathrm{H}_{G}(X)$, then the induced map\n\\[\ng^*g_*:\\mathrm{H}_{c,G}(X,\\phi_f)^*\\rightarrow \\mathrm{H}_{c,G}(X,\\phi_f)^*[\\eu(g)^{-1}]\n\\]\nis given by division by $\\eu(g)$.\n\n\n\\end{itemize}\n\\end{proposition}\n\n\\section{Cohomological Hall algebra}\n\\label{coha_section}\n\\subsection{Spaces of quiver representations} \\label{Hdef}\nLet $Q$ be a quiver, as in the introduction. We denote by $Q_0$ the vertices of $Q$, and by $Q_1$ the arrows, so that there are two maps $s,t:Q_1\\rightarrow Q_0$ taking an arrow to its source and target. Let $\\mathcal{C}$ be a symmetric tensor category with monoidal product $\\boxtimes$. We assume also that $\\mathcal{C}$ is equipped with an invertible shift functor $\\state{1\/2}$ and natural isomorphisms of bifunctors $\\state{1\/2}\\circ \\boxtimes\\cong \\boxtimes\\circ(\\id\\otimes \\state{1\/2})\\cong\\boxtimes\\circ(\\state{1\/2}\\otimes \\id)$. In all our examples this is achieved in a canonical way by first defining $\\mathbf{1}_{\\mathcal{C}}\\state{1\/2}$ and then defining $M\\state{1\/2}:=M\\boxtimes \\mathbf{1}_{\\mathcal{C}}\\state{1\/2}$. We define $\\mathcal{C}_Q$ to be the category with objects the $\\mathbb{Z}^{Q_0}$-graded objects in $\\mathcal{C}$, i.e. objects $\\mathcal{L}$ in $\\mathcal{C}$ equipped with a decomposition\n\\[\n\\mathcal{L}\\cong\\bigoplus_{\\gamma\\in\\mathbb{Z}^{Q_0}}\\mathcal{L}_{\\gamma}.\n\\]\nThe morphisms in $\\mathcal{C}_Q$ are the morphisms of the underlying objects in $\\mathcal{C}$ that respect the decompositions. We make $\\mathcal{C}_Q$ a tensor category by setting\n\\[\n\\bigoplus_{\\gamma_1\\in\\mathbb{Z}^{Q_0}}\\mathcal{L}_{\\gamma_1}\\boxtimes_+^{\\tw}\\bigoplus_{\\gamma_2\\in\\mathbb{Z}^{Q_0}}\\mathcal{L}'_{\\gamma_2}:=\\bigoplus_{\\gamma\\in\\mathbb{Z}^{Q_0}}\\left(\\bigoplus_{\\gamma_1+\\gamma_2=\\gamma} \\mathcal{L}_{\\gamma_1}\\otimes\\mathcal{L}'_{\\gamma_2}\\right)\\state{\\chi(\\gamma_1,\\gamma_2)\/2-\\chi(\\gamma_2,\\gamma_1)\/2}.\n\\]\nThis monoidal structure is not isomorphic to the symmetric monoidal structure defined by\n\\begin{equation}\n\\label{symmMon}\n\\bigoplus_{\\gamma_1\\in\\mathbb{Z}^{Q_0}}\\mathcal{L}_{\\gamma_1}\\boxtimes_+\\bigoplus_{\\gamma_2\\in\\mathbb{Z}^{Q_0}}\\mathcal{L}'_{\\gamma_2}:=\\bigoplus_{\\gamma\\in\\mathbb{Z}^{Q_0}}\\left(\\bigoplus_{\\gamma_1+\\gamma_2=\\gamma} \\mathcal{L}_{\\gamma_1}\\otimes\\mathcal{L}'_{\\gamma_2}\\right),\n\\end{equation}\nindeed the monoidal structure $\\boxtimes_{+}^{\\tw}$ can not in general even be made into a braided monoidal structure, while the monoidal structure (\\ref{symmMon}) extends to a symmetric monoidal structure.\n\\smallbreak\nLet $W\\in \\mathbb{C} Q\/[\\mathbb{C} Q,\\mathbb{C} Q]$ be a potential for $Q$. We call a pair $(Q,W)$ a QP from now on. For $\\gamma\\in\\mathbb{N}^{Q_0}$ denote by $M_{Q,\\gamma}$ the affine space\n\\[\n\\bigoplus_{a\\in Q_1}\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{\\gamma(s(s))},\\mathbb{C}^{\\gamma(t(a))}).\n\\]\nThis is considered in a natural way as a space of representations of $Q$, or a space of left $\\mathbb{C} Q$-modules. If $\\gamma_1,\\gamma_2\\in\\mathbb{N}^{Q_0}$ is a pair of dimension vectors, we denote by $M_{Q,\\gamma_1,\\gamma_2}$ the affine space\n\\[\n\\bigoplus_{a\\in Q_1}\\big\\{f_a\\in\\mathop{\\rm Hom}\\nolimits\\big(\\mathbb{C}^{\\gamma_1(s(a))}\\oplus\\mathbb{C}^{\\gamma_2(s(a))},\\mathbb{C}^{\\gamma_1(t(a))}\\oplus\\mathbb{C}^{\\gamma_2(t(a))}\\big)|f_a(\\mathbb{C}^{\\gamma_1(s(a))})\\subset\\mathbb{C}^{\\gamma_1(t(a))}\\big\\}.\n\\]\nIf $\\gamma_1+\\gamma_2=\\gamma$, there is a natural inclusion $\\eta:M_{Q,\\gamma_1,\\gamma_2}\\hookrightarrow M_{Q,\\gamma}$. In what follows, if the quiver $Q$ is fixed, we will abbreviate $M_{Q,\\gamma}$ to $M_{\\gamma}$ and abbreviate $M_{Q,\\gamma_1,\\gamma_2}$ to $M_{\\gamma_1,\\gamma_2}$.\n\\smallbreak\nDefine $G_{\\gamma}:=\\prod_{i\\in Q_0}\\Gl_{\\mathbb{C}}(\\gamma(i))$ and $G_{\\gamma_1,\\gamma_2}:=\\prod_{i\\in Q_0}\\Gl_{\\mathbb{C}}(\\gamma_1(i),\\gamma_2(i))$, where $\\Gl_{\\mathbb{C}}(m,n)$ is the subgroup of $\\Gl_{\\mathbb{C}}(m+n)$ preserving the flag $0\\subset \\mathbb{C}^m\\subset\\mathbb{C}^{m+n}$. On each $M_{\\gamma}$ there is a function $\\tr(W)_{\\gamma}$, which is invariant with respect to the action of $G_{\\gamma}$.\n\\smallbreak\nWe define in the same way the function $\\tr(W)_{\\gamma_1,\\gamma_2}$ on $M_{\\gamma_1,\\gamma_2}$, which is again invariant with respect to the $G_{\\gamma_1,\\gamma_2}$-action. There is a natural projection $p:M_{\\gamma_1,\\gamma_2}\\rightarrow M_{\\gamma_1}\\times M_{\\gamma_2}$ and an inclusion $\\eta:M_{\\gamma_1,\\gamma_2}\\rightarrow M_{\\gamma_1+\\gamma_2}$ and we have \n\\begin{align*}\n\\tr(W)_{\\gamma_1,\\gamma_2}=&p^*(\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2})\\\\=&\\eta^*\\tr(W)_{\\gamma_1+\\gamma_2}.\n\\end{align*}\nIn what follows we will use the symbol $\\Sp$ to denote a property of $\\mathbb{C}Q$-modules, stable under isomorphisms of representations. Since points of $M_{\\gamma}$ represent $\\mathbb{C}Q$-modules, we can define subsets $M^{\\Sp}_{\\gamma}\\subset M_{\\gamma}$ as those subsets of representations satisfying property $\\Sp$. We will always pick $\\Sp$ so that this inclusion is the inclusion of closed points induced by an inclusion of algebraic varieties. Furthermore, since we assume that $\\Sp$ is stable under isomorphism, it will follow that $M_{\\gamma}^{\\Sp}\\subset M_{\\gamma}$ is an inclusion of $G_{\\gamma}$-equivariant varieties.\n\\begin{assumption}\n\\label{closed_under}\nThe subspaces $M^{\\Sp}_{\\gamma}$ are required to satisfy the property that there is an equality for all pairs $\\gamma_1,\\gamma_2\\in\\mathbb{N}^{Q_0}$\n\\begin{align*}\n&M_{\\gamma_1,\\gamma_2}^{\\Sp,\\mathop{\\rm ext}\\nolimits}:=p^{-1}(M_{\\gamma_1}^{\\Sp}\\times M_{\\gamma_2}^{\\Sp})=\\\\& M^{\\Sp}_{\\gamma_1,\\gamma_2}:=\\eta^{-1} (M_{\\gamma_1+\\gamma_2}^{\\Sp}).\n\\end{align*}\n\\end{assumption}\n\n\nWe will also assume that $0\\in M_{0}^{\\Sp}$, which is equivalent to not all of the $M^{\\Sp}_{\\gamma}$ being empty. \n\\smallbreak\n\n\\begin{remark}\nFor convenience we assume that $M_{\\gamma}^{\\Sp}\\cap \\crit(\\tr(W)_{\\gamma})\\subset \\tr(W)_{\\gamma}^{-1}(0)$ for all $\\gamma\\in\\mathbb{N}^{Q_0}$. This last requirement can be relaxed at the expense of some extra minor complications, but none of the applications we are interested in will require this.\n\\end{remark}\n\\begin{example}\nLet $\\zeta\\in\\mathbb{H}_+^{Q_0}:=\\{re^{i\\theta}|r\\in\\mathbb{R}_{>0},\\theta\\in(0,\\pi]\\}^{Q_0}$. We call such a $\\zeta$ a stability condition for $Q$. A representation $\\rho$ of $Q$ is called $\\zeta$-semistable if for all nonzero subrepresentations $\\rho'\\subset \\rho$, $\\arg(\\zeta\\cdot\\dim(\\rho'))\\leq \\arg(\\zeta\\cdot \\dim(\\rho))$, and $\\rho$ is called $\\zeta$-stable if this inequality is strict for all proper $\\rho'\\subset\\rho$. Fix $\\theta\\in (0,\\pi]$. One can easily check that the condition on a $\\mathbb{C}Q$-module $M$ of being $\\zeta$-semistable and satisfying $\\arg(\\zeta\\cdot\\dim(M))=\\theta$ satisfies Assumption \\ref{closed_under}. The property of being $\\zeta$-stable with fixed $\\theta$ does not satisfy Assumption \\ref{closed_under} (stability is not preserved under taking direct sums, for instance). So we may define the (underlying object of the) cohomological Hall algebra of $\\zeta$-semistable $\\mathbb{C}Q$-modules of slope $\\theta$ with potential $W$:\n\\[\n\\mathcal{H}^{\\zeta-ss}_{Q,W,\\theta}:=\\bigoplus_{\\gamma\\in\\mathbb{N}^{Q_0}|\\arg(\\zeta\\cdot\\gamma)=\\theta}\\mathrm{H}_{c,G_{\\gamma}}(M^{\\zeta-ss}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\\state{{-\\chi(\\gamma,\\gamma)\/2}}^*.\n\\]\nHere, since $M^{\\zeta-ss}_{\\gamma}$ is an open subscheme of $M_{\\gamma}$, it does not matter whether we take vanishing cycles on $M^{\\zeta-ss}_{\\gamma}$ of the restriction of the function $\\tr(W)_{\\gamma}$, or instead take the restriction to $M^{\\zeta-ss}_{\\gamma}$ of the vanishing cycles complex $\\phi_{\\tr(W)_{\\gamma}}$. However, for general $\\Sp$ these two objects are \\textit{not} the same, and we consider the latter.\n\\end{example}\n\nWe define \n\\begin{equation}\n\\label{uldef}\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}:=\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{{\\chi(\\gamma,\\gamma)\/2}}},\n\\end{equation}\nwhere the twist is as defined in (\\ref{statedef}), and define\n\\[\n\\mathcal{H}^{\\Sp}_{Q,W}:=\\bigoplus_{\\gamma\\in\\mathbb{Z}^{Q_0}}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}.\n\\]\nIn the case in which all $M^{\\Sp}_{Q,W,\\gamma}=M_{Q,W,\\gamma}$ we denote $\\mathcal{H}^{\\Sp}_{Q,W}$ by $\\mathcal{H}_{Q,W}$.\n\nWe consider (\\ref{uldef}) as an object in $\\mathcal{C}_Q$, where $\\mathcal{C}$ can be, for example, each of the following choices of symmetric tensor categories.\n\\begin{itemize}\n\\item\n$\\mathcal{C}$ can be taken to be $\\Db{\\Vect}$. In this case we lose all Hodge theoretic information regarding vanishing cycles, and only consider the underlying vector spaces in (\\ref{uldef}). The shift functor $\\{1\/2\\}$ is then the usual cohomological shift functor $[1]$.\n\\item\n$\\mathcal{C}$ can be taken to be $\\Db{\\mathop{\\mathbf{MHS}}\\nolimits}$. In this case we remember much of the mixed Hodge structure underlying (\\ref{uldef}), but some care has to be taken when we try to turn (\\ref{uldef}) into an algebra, as the Thom--Sebastiani isomorphism, which forms part of the definition of the multiplication map, does not necessarily respect mixed Hodge structures. For a class of examples in which the Thom--Sebastiani isomorphism does respect the underlying mixed Hodge structures, see the appendix, and especially Proposition \\ref{TScomm}. As a further complication, the shift $\\state{{-}}$ has no satisfactory definition in $\\Db{\\mathop{\\mathbf{MHS}}\\nolimits}$; this category has to be slightly extended as in \\cite[Sec.3.4]{COHA}.\n\\item\n$\\mathcal{C}$ can be taken to be $\\Db{\\mathop{\\mathbf{MMHS}}\\nolimits}$. In this case the shift functor $\\state{{-1\/2}}$ is given by tensoring with $\\mathrm{H}_c(\\mathbb{A}^1,\\phi_{x^2})$.\n\\item\nDefine $\\mathbb{Z}_{{\\rm wt}}:=\\mathbb{Z}$. Then $\\mathcal{C}$ can be taken to be $\\Db{\\Vect_{\\mathbb{Z}_{{\\rm wt}}}}$, the category of graded vector spaces, but in this case we should replace $\\mathcal{H}_{Q,W}^{\\Sp}$ by its associated graded object with respect to the weight filtration on the underlying monodromic mixed Hodge structure. The shift functor $\\state{1\/2}$ is given by tensoring with $\\mathbb{Q}$, given bidegree $(-1,-1)$ with respect to the cohomological and weight grading.\n\\end{itemize}\n\n\n\\begin{remark}\n\\label{JacRem}\nAs an aside we explain the representation-theoretic origin of $\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})$. Let $S$ be the set of cyclic paths in $Q$. Given a quiver $Q$, and a potential $W=\\sum_{m\\in S} \\lambda_m m\\in \\mathbb{C} Q\/[\\mathbb{C} Q,\\mathbb{C} Q]$, where each of the $\\lambda_m$ is a scalar, and all but finitely many of the $\\lambda_m$ are zero, the noncommutative derivative $\\partial W\/\\partial a$ is defined by setting\n\\[\n\\partial W\/\\partial a:=\\sum_{m\\in S}\\lambda_m\\sum_{m=uav|u,v\\text{ paths in }Q}vu.\n\\]\nThen the Jacobi algebra for the QP $(Q,W)$ is defined by\n\\[\nJ(Q,W):=\\mathbb{C} Q\/\\langle \\partial W\/\\partial a|a\\in Q_1\\rangle.\n\\]\nRepresentations of the Jacobi algebra form a Zariski closed subscheme $\\mathcal{V}_{\\gamma}$ of $M_{Q,\\gamma}$ in the natural way, and we have the equality of subschemes\n\\[\n\\mathcal{V}_{\\gamma}=\\crit(\\tr(W)_{\\gamma}).\n\\]\nIn addition, the vanishing cycle complex $\\phi_{\\tr(W)_{\\gamma}}$ is supported on the critical locus of $\\tr(W)_{\\gamma}$. So the compactly supported equivariant cohomology $\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})$ can be thought of as the compactly supported equivariant cohomology of the stack of $\\gamma$-dimensional representations of $J(Q,W)$ satisfying property $\\Sp$, with coefficients in the vanishing cycle complex.\n\\end{remark}\n\n\n\\subsection{Multiplication for $\\mathcal{H}^{\\Sp}_{Q,W}$} \\label{MOsec}In this subsection we recall the associative product of \\cite[Sec.7.6]{COHA}\n\\[\nm:\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}\\boxtimes^{\\tw}_+\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}\\rightarrow \\mathcal{H}^{\\Sp}_{Q,W,\\gamma}.\n\\]\nAs above, all functors will be considered as derived functors.\n\nConsider the affine fibrations \n\\begin{equation}\n\\label{affpf}\np_{\\gamma_1,\\gamma_2,N}:\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N\\rightarrow \\overline{(M_{\\gamma_1}\\times M_{\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N.\n\\end{equation}\nThese induce isomorphisms\n\\[\n\\phi_{(\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2})_N}\\left(\\mathbb{Q}_{\\overline{(M_{\\gamma_1}\\times M_{\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\rightarrow p_{\\gamma_1,\\gamma_2,N,*}\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\right),\n\\]\nand via Verdier duality, morphisms\n\\begin{equation}\n\\phi_{(\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2})_N}\\left(p_{\\gamma_1,\\gamma_2,N,!}D\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\rightarrow D\\mathbb{Q}_{\\overline{(M_{\\gamma_1}\\times M_{\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\right),\n\\end{equation}\nor\n\\begin{equation}\n\\phi_{(\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2})_N}\\left(p_{\\gamma_1,\\gamma_2,N,!}\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\rightarrow \\mathbb{Q}_{\\overline{(M_{\\gamma_1}\\times M_{\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\state{{-l_1(\\gamma_2,\\gamma_1)}}\\right),\n\\end{equation}\nusing (\\ref{CanDu}). The function $l_1$ is as defined in equations (\\ref{ldefs}).\nBy Corollary \\ref{ppoutside} we obtain isomorphisms\n\\begin{align}\n&p_{\\gamma_1,\\gamma_2,N,!}\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2,N}}(\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N})\\rightarrow \\\\&\\nonumber\\phi_{(\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2})_N}(\\mathbb{Q}_{\\overline{(M_{\\gamma_1}\\times M_{\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N}\\state{{-l_1(\\gamma_2,\\gamma_1)}}).\n\\end{align}\nApplying the shift functor, passing to the limit and take compactly supported cohomology, we arrive at an isomorphism\n\\begin{align}\n(\\alpha)^*:&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})\\state{{-\\chi(\\gamma,\\gamma)\/2+l_0(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\ \\nonumber &\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1}^{\\Sp}\\times M_{\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}})\\state{{-\\chi(\\gamma,\\gamma)\/2+\\chi(\\gamma_2,\\gamma_1)}},\n\\end{align}\nthough we work with the dual map\n\\begin{align}\n\\alpha:&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1}^{\\Sp}\\times M_{\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\ \\nonumber &\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-l_0(\\gamma_2,\\gamma_1)}},\n\\end{align}\nas $\\mathcal{H}_{Q,W}^{\\Sp}$ is defined in terms of dual spaces of compactly supported equivariant cohomology. In the terminology of Section \\ref{umkehr_sec}, these maps are induced by the pullback maps associated to the maps (\\ref{affpf}).\n\\smallbreak\nBy the definitions of $\\chi,l_0,l_1$ we have the equality\n\\begin{align*}\n&\\chi(\\gamma,\\gamma)\/2-l_0(\\gamma_2,\\gamma_1)+l_1(\\gamma_2,\\gamma_1)=\n\\\\\n&\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)=\n\\\\\n&\\chi(\\gamma_1,\\gamma_1)\/2+\\chi(\\gamma_2,\\gamma_2)\/2+\\chi(\\gamma_1,\\gamma_2)\/2-\\chi(\\gamma_2,\\gamma_1)\/2,\n\\end{align*}\nthe degree shift in the definition of $\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_{+}^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_2}$.\n\\smallbreak\n\nSimilarly, the affine fibrations \n\\[\nq_{\\gamma_1,\\gamma_2,N}:\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N\\rightarrow \\overline{(M_{\\gamma_1,\\gamma_2}, G_{\\gamma_1,\\gamma_2})}_N\n\\]\ninduce maps\n\\begin{equation}\n\\label{af2}\nq_{\\gamma_1,\\gamma_2,N,!}\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2,N}}(\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1}\\times G_{\\gamma_2})}_N})\\rightarrow \\phi_{\\tr(W)_{\\gamma_1,\\gamma_2,N}}(\\mathbb{Q}_{\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1,\\gamma_2})}_N})\\state{{-l_0(\\gamma_1,\\gamma_2)}}.\n\\end{equation}\nTaking shifted compactly supported cohomology and taking duals we obtain pullback isomorphisms\n\\begin{align}\n\\label{betaMap}\n\\beta:&\\mathrm{H}_{c, G_{\\gamma_1,\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2}}\\rightarrow \\\\&\\mathrm{H}_{c, G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-l_0(\\gamma_2,\\gamma_1)}}.\\nonumber\n\\end{align}\nConsider the proper maps\n\\begin{equation}\n\\label{prdef}\npr_{\\gamma_1,\\gamma_2,N}:\\overline{(M_{\\gamma},G_{\\gamma_1,\\gamma_2}})_N\\rightarrow \\overline{(M_{\\gamma},G_{\\gamma}})_N. \n\\end{equation}\nThe natural transformation of functors\n\\[\n\\phi_{\\tr(W)_{\\gamma,N}}\\rightarrow pr_{\\gamma_1,\\gamma_2,N,!}\\phi_{\\tr(W)_{\\gamma,N}}pr_{\\gamma_1,\\gamma_2,N}^*\n\\]\napplied to $\\mathbb{Q}_{\\overline{(M_{\\gamma},G_{\\gamma})}_N}$ induces a map\n\\begin{equation}\n(\\delta)^*:\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\\rightarrow \\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\n\\end{equation}\nor, taking shifted duals\n\\begin{align}\n\\label{pfdef}\n\\delta:&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\rightarrow\\\\& \\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}},\\nonumber\n\\end{align}\nthe pushforward map associated to the maps (\\ref{prdef}). Here, in passing to the limit, we have applied Proposition \\ref{mixingprop}.\n\n\nNext, consider the inclusions $i_{\\gamma_1,\\gamma_2,N}:\\overline{(M_{\\gamma_1,\\gamma_2},G_{\\gamma_1,\\gamma_2})}_N\\rightarrow \\overline{(M_{\\gamma},G_{\\gamma_1,\\gamma_2})}_N$. These induce maps\n\\begin{equation}\n\\label{zetapd}\n\\phi_{\\tr(W)_{\\gamma,N}}\\rightarrow i_{\\gamma_1,\\gamma_2,N,*}i_{\\gamma_1,\\gamma_2,N}^*\\phi_{\\tr(W)_{\\gamma,N}}\n\\end{equation}\nand maps\n\\[\n(\\zeta)^*:\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\\rightarrow\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma}})\n\\]\nor, again taking shifted duals,\n\\begin{align}\n\\zeta:&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}.\n\\end{align}\n\nFinally, the natural transformation\n\\[\n\\phi_{\\tr(W)_{\\gamma,N}}\\rightarrow i_{\\gamma_1,\\gamma_2,N,*}\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2,N}}i_{\\gamma_1,\\gamma_2,N}^*\n\\]\ninduces maps\n\\begin{align}\n\\label{epsdef}\n\\epsilon:&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\nonumber\n\\end{align}\nas in the definition of $\\delta$.\n\n\nThroughout we have used that $pr_{\\gamma_1,\\gamma_2,N}$ and $i_{\\gamma_1,\\gamma_2,N}$ are proper in order to identify $pr_{\\gamma_1,\\gamma_2,N,*}$ with $pr_{\\gamma_1,\\gamma_2,N,!}$ and $i_{\\gamma_1,\\gamma_2,N,*}$ with $i_{\\gamma_1,\\gamma_2,N,!}$. The passage to the limit in the definition of $\\zeta\\epsilon$ is again an application of Proposition \\ref{mixingprop}.\n\nWe define a product \n\\[\nm:\\mathcal{H}^{\\Sp}_{Q,W}\\boxtimes_+^{\\tw} \\mathcal{H}^{\\Sp}_{Q,W}\\rightarrow \\mathcal{H}^{\\Sp}_{Q,W}\n\\]\nto be the map which, when restricted to $\\mathcal{H}_{Q,W,\\gamma_1}^{\\Sp}\\boxtimes_+^{\\tw}\\mathcal{H}_{Q,W,\\gamma_2}^{\\Sp}$, is given by the composition of maps $\\delta\\zeta\\epsilon\\beta^{-1}\\alpha\\TS$, where $\\TS$ is the shift of the Thom--Sebastiani isomorphism\n\\begin{align*}\n\\mathrm{H}_{c,G_{\\gamma_1}}(M^{\\Sp}_{\\gamma_1},\\phi_{\\tr(W)_{\\gamma_1}})^*\\otimes \\mathrm{H}_{c,G_{\\gamma_2}}(M^{\\Sp}_{\\gamma_2},\\phi_{\\tr(W)_{\\gamma_2}})^*\\rightarrow&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M^{\\Sp}_{\\gamma_1}\\times M^{\\Sp}_{\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}})^*.\n\\end{align*}\n\n\\begin{proposition}[\\cite{COHA}]\n\\label{IntId}\nThe map $\\epsilon$ is an isomorphism.\n\\end{proposition}\nProposition \\ref{IntId} is the Hodge theoretic version of the ``integral identity'' from \\cite{KS}, and is proved as Theorem 13 of \\cite{COHA}. In fact the result will not be used in this paper, its main purpose is to show that the integration map of \\cite[Sec.6.3]{KS} is a ring homomorphism, but we will not be concerned with the integration map.\n\\begin{remark}\nThe map $\\zeta\\epsilon$ is the shifted pushforward \n\\begin{align*}\n\\underline{\\zeta}:&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\&\n\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\n\\end{align*}\nassociated to $i_{\\gamma_1,\\gamma_2}$. We have split it into two maps as in \\cite{COHA}. In subsequent sections we will just use $\\underline{\\zeta}$.\n\\end{remark}\nFor the benefit of the reader we represent the multiplication in a different way:\n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma_1+\\gamma_2}}(M^{\\Sp}_{\\gamma_1+\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1+\\gamma_2}})^*\n&\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(1,1){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(1,1){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(1,1){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigr)^*\\ar[l]^-{=:}\n\\\\\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,1){\\circle*{3}}\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigr)^*\\ar[r]^-{\\zeta}\n&\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,1){\\circle*{3}}\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(1,1){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigr)^*\\ar[u]^-{\\delta}\n\\\\\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigr)^*\\ar[u]^{\\epsilon}\\ar[ur]^{\\underline{\\zeta}}\n&\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\\put(5,5){\\circle*{3}}\n\\end{picture}}\\bigr)\\state{-l_0(\\gamma_1,\\gamma_2)}^*\\ar[l]_-{\\beta^{-1}}\n\\\\\n\\left(\\mathrm{H}_{c,G_{\\gamma_1}}(M^{\\Sp}_{\\gamma_1},\\phi_{\\tr(W)_{\\gamma_1}})\\otimes\\mathrm{H}_{c,G_{\\gamma_1}}(M^{\\Sp}_{\\gamma_2},\\phi_{\\tr(W)_{\\gamma_2}})\\right)^*\\ar[r]_-{\\TS}\n&\n\\mathrm{H}_{\\begin{picture}(7,9)\n\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\n\\end{picture}}\\bigl(M_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\n\\end{picture}},W_{\\begin{picture}(7,9)\\put(1,5){\\circle*{3}}\\put(5,1){\\circle*{3}}\n\\end{picture}}\\bigr)^*\\ar[u]_-{\\alpha}\n}\n\\]\nwhere we have abbreviated the notation in the obvious ways, and left out the shifts.\n\\smallbreak\n\nWe work with the nonsymmetric monoidal structure $\\boxtimes_+^{\\tw}$ in order for $m$ to preserve cohomological degree, and for $m$ to be an unshifted map in $\\mathcal{C}_Q$, e.g. to avoid the appearance of any Tate twists in the definition of $m$, assuming that our background category $\\mathcal{C}$ is set to be $\\mathop{\\mathbf{MMHS}}\\nolimits$, the category of mondromic mixed Hodge structures.\n\\begin{definition}\n\\label{degpres}\nWe say that $\\Sp$ is \\textit{degree preserving} if it satisfies the following condition: for all $\\gamma_1$ and $\\gamma_2$ such that $M^{\\Sp}_{\\gamma_1}$ and $M^{\\Sp}_{\\gamma_2}$ are non-empty, $\\chi(\\gamma_1,\\gamma_2)=\\chi(\\gamma_2,\\gamma_1)$.\n\\end{definition}\n\\begin{proposition}\nLet $\\Sp$ be degree-preserving. Then $(\\mathcal{H}^{\\Sp}_{Q,W},m)$ is an algebra object in the category $\\mathcal{C}_Q$ with the untwisted symmetric monoidal product $\\boxtimes_+$.\n\\end{proposition}\n\\begin{example}\n\\label{symmEx}\nAssume that $Q$ is symmetric. Then $\\chi(\\bullet,\\bullet)$ is symmetric, and so every property $\\Sp$ is degree-preserving, and so $\\mathcal{H}^{\\Sp}_{Q,W}$ is an algebra in the category $\\mathcal{C}_Q$ with the symmetric monoidal product $\\boxtimes_+$ for every property $\\Sp$.\n\\end{example}\n\\begin{example}\nAssume that the stability condition $\\zeta$ is generic, in the sense that $\\arg(\\zeta\\cdot\\gamma_1)=\\arg(\\zeta\\cdot\\gamma_2)$ implies that $\\gamma_1=r\\gamma_2$ for some $r\\in\\mathbb{R}$. Then setting $\\Sp$ to be the property that a representation $M$ of $\\mathbb{C}Q$ is $\\zeta$-semistable, and $\\arg(\\zeta\\cdot \\dim(M))=\\theta$ for some fixed $\\theta$, the property $\\Sp$ is degree preserving. In fact the requirement that $\\Sp$, so defined, is degree-preserving, is often the requirement that one is most interested in guaranteeing by imposing genericity so elsewhere (see for example \\cite{MeRe14}), genericity for $\\zeta$ is just defined by the weak requirement that $\\zeta$ be degree-preserving. By Example \\ref{symmEx}, then, all stability conditions on a symmetric quiver may be considered to be generic.\n\\end{example}\n\n\n\\section{The critical CoHA as a shuffle algebra}\n\\label{t_sec}\n\n\\subsection{Localisation} Let $\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_r}$ be an arbitrary finite tensor product of $\\mathbb{N}^{Q_0}$-graded pieces of the CoHA $\\mathcal{H}_{Q,W}^{\\Sp}$. Via the Thom--Sebastiani isomorphism we may identify the vector spaces\n\\begin{equation}\n\\label{bigTens}\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_r}\\cong \\mathrm{H}_{c,G_{\\gamma_1}\\times\\ldots\\times G_{\\gamma_r}}(M^{\\Sp}_{\\gamma_1}\\times\\ldots\\times M^{\\Sp}_{\\gamma_r},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\ldots\\boxplus\\tr(W)_{\\gamma_r}})^*,\n\\end{equation}\nwhere we have left out the shift in cohomology. As detailed in Section \\ref{moduleStruc}, the left hand side of (\\ref{bigTens}) carries an action of\n\\[\nA_{\\gamma_1,\\ldots,\\gamma_r}:=\\bigotimes_{c=1,\\ldots,r}\\left(\\bigotimes_{i\\in Q_0}\\mathbb{Q}\\left[x^{(c)}_{i,1},\\ldots,x^{(c)}_{i,\\gamma_c(i)}\\right]\\right)^{{\\rm Sym}_{\\gamma_c}}.\n\\]\nLet $M:=M^{\\Sp}_{\\gamma_1}\\times\\ldots\\times M^{\\Sp}_{\\gamma_r}$ and $G:=G_{\\gamma_1}\\times\\ldots\\times G_{\\gamma_r}$. Let $c,d\\in\\{1,\\ldots,r\\}$ be a pair of distinct numbers, and let $i,i'\\in Q_0$. Let\n\\[\nT=M\\times\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{\\gamma_c(i)},\\mathbb{C}^{\\gamma_d(i')}).\n\\]\nThen $T$ is naturally a $G$-equivariant vector bundle over $M$, with projection $\\pi:T\\rightarrow M$, section $M\\xrightarrow{z\\mapsto (z,0)} M\\times \\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{\\gamma_c(i)},\\mathbb{C}^{\\gamma_d(i')})$ and an Euler class defined as in Section \\ref{umkehr_sec}, which we denote $\\eu(i,i',c,d)$.\n\\begin{proposition}\n\\label{ABprop}\nMultiplication by $\\eu(i,i',c,d)$ is an injective endomorphism of $\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_r}$.\n\\end{proposition}\n\\begin{proof}\nThis is a small variation of the Atiyah Bott lemma, proved in \\cite{AtBo83}. In detail, let $S^1\\rightarrow G_{\\gamma_c}$ be defined by $e^{\\sqrt{-1}\\theta}\\mapsto e^{\\sqrt{-1}\\theta}\\id$. Via the inclusion of algebraic groups $G_{\\gamma_c}\\rightarrow G_{\\gamma_1}\\times\\ldots\\times G_{\\gamma_r}$ this defines an action on the total space $T$ such that the fixed point set is exactly $M$. Let $G'=G\/S^1$. Then $\\mathrm{H}_{c,G}(M^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\ldots\\boxplus\\tr(W)_{\\gamma_r}})^*$ is filtered by \n\\[\nF^p\\left(\\mathrm{H}_{c,G}(M^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\ldots\\boxplus\\tr(W)_{\\gamma_r}})^*\\right):=\\mathrm{H}^{\\geq p}_{c,G'}(M^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\ldots\\boxplus\\tr(W)_{\\gamma_r}})^*\\otimes \\mathrm{H}_{S^1}(\\mathop{\\rm pt},\\mathbb{Q}), \n\\]\nand we denote by $N$ the associated graded object, which is acted on freely by $\\mathrm{H}_{S^1}(\\mathop{\\rm pt},\\mathbb{Q})$. Let \n\\[\n\\tilde{\\mu}\\in \\mathrm{H}_{c,G}(M^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\ldots\\boxplus\\tr(W)_{\\gamma_r}})^*, \n\\]\nand let $\\mu\\in N$ be the associated homogeneous element. Let $s$ equal the degree of $\\mu$ with respect to the grading induced by $F$. Then projecting $\\eu(i,i',c,d)\\mu$ onto its degree $s$ part, also with respect to the grading induced by $F$, it is given by $\\eu_{S^1}(i,i',c,d)\\mu$, where now $\\eu_{S^1}(i,i',c,d)$ is the $S^1$-equivariant Euler characteristic of $T$, which is nonzero since $M$ is the fixed locus of the $S^1$-action on $T$.\n\\end{proof}\n\\begin{definition}\nWe define \n\\[\n\\eue(Q_1,\\gamma_c,\\gamma_d)=\\prod_{a\\in Q_1}\\prod_{m=1}^{\\gamma_c(s(a))}\\nolimits\\prod_{m'=1}^{\\gamma_d(t(a))}\\nolimits(x_{t(a),m'}^{(d)}-x^{(c)}_{s(a),m})\n\\]\nand\n\\[\n\\eue(Q_0,\\gamma_c,\\gamma_d)=\\prod_{i\\in Q_0}\\prod_{m=1}^{\\gamma_c(i)}\\nolimits\\prod_{m'=1}^{\\gamma_d(i)}\\nolimits(x_{i,m'}^{(d)}-x^{(c)}_{i,m}).\n\\]\n\\end{definition}\nEach of these classes is a product of classes of the form $\\eu(i,i',c,d)$, and so we deduce from Proposition \\ref{ABprop} that multiplication by $\\eue(Q_\\iota,\\gamma_c,\\gamma_d)$ is an injective endomorphism of $\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_c}\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_d}$ for $\\iota=0,1$.\n\\subsection{Definition of the $T$-equivariant multiplication} \n\\label{TCOHA}\nDefine $T_{\\gamma}:=\\prod_{i\\in Q_0} (\\mathbb{C}^*)^{\\gamma(i)}$. After choosing an ordered basis for $\\bigoplus_{i\\in Q_0} \\mathbb{C}^{\\gamma(i)}$ there is a natural inclusion $T_{\\gamma}\\subset G_{\\gamma}$; we consider the natural ordered basis. Let $\\No(T_{\\gamma})$ be the normalizer of $T_{\\gamma}$ inside $G_{\\gamma}$. For every natural number $N$ there are morphisms\n\\begin{equation}\n\\label{nin}\ns_N:\\overline{(M_{Q,\\gamma},\\No(T_{\\gamma}))}_N\\rightarrow \\overline{(M_{Q,\\gamma},G_{\\gamma})}_N\n\\end{equation}\ninduced by the inclusion $\\No(T_{\\gamma})\\subset G_{\\gamma}$. As with equation (\\ref{phibs}), the maps (\\ref{nin}) induce maps \n\\begin{equation}\n\\label{aai}\n\\phi_{\\tr(W)_{\\gamma,N}}\\rightarrow s_{N,*}\\phi_{\\tr(W)_{\\gamma,N}}\n\\end{equation}\nand\n\\[\ns_{N,!}\\phi_{\\tr(W)_{\\gamma,N}}\\state{{l(\\gamma)}}\\rightarrow \\phi_{\\tr(W)_{\\gamma,N}}\n\\]\nwhere \n\\[\nl(\\gamma):=\\sum_{i\\in Q_0}(\\gamma_i^2-\\gamma_i), \n\\]\nand we abuse notation by denoting by $\\tr(W)_{\\gamma, N}$ the functions defined by $\\tr(W)_{\\gamma}$ on $\\overline{(M_{Q,\\gamma},\\No(T_{\\gamma}))}_N$ as well as on $\\overline{(M_{Q,\\gamma},G_{\\gamma})}_N$.\n\\begin{proposition}\n\\label{invprop}\nThere are natural maps\n\\begin{equation}\n\\label{invpart}\n\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma}}\\state{{l(\\gamma)}}\\rightarrow\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\n\\end{equation}\nwhich are isomorphisms.\n\\end{proposition}\n\\begin{proof}\nFor each $N$, there is a Galois cover\n\\[\nw:\\overline{(M^{\\Sp}_{\\gamma},T_{\\gamma})}_N\\rightarrow \\overline{(M^{\\Sp}_{\\gamma},\\No(T_{\\gamma}))}_N\n\\]\nwith Galois group $\\SG_{\\gamma}$, from which we deduce that $\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},w^*\\phi_{\\tr(W)_{\\gamma}})$ carries a $\\SG_{\\gamma}$-action and there is an isomorphism in compactly supported cohomology\n\\[\n\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},w^*\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma}}\\rightarrow \\mathrm{H}_{c,\\No(T_{\\gamma})}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}}).\n\\]\nSo it suffices to prove that \n\\[\n\\mathrm{H}_{c,\\No(T_{\\gamma})}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\\state{{l(\\gamma)}}\\rightarrow\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})\n\\]\nis an isomorphism. This will follow from the claim that (\\ref{aai}) is an isomorphism in $\\Dbc{\\overline{(M_{\\gamma}^{\\Sp},G_{\\gamma})}_N}$. Since $s_N$ is smooth, it follows that \n\\[\n\\phi_{\\tr(W)_{\\gamma,N}\\circ s_N}\\cong s_N^*\\phi_{\\tr(W)_{\\gamma,N}},\n\\]\nand so the proof follows from the claim that if $\\mathcal{F}$ is an object of $\\Dbc{\\overline{(M_Q,G_{\\gamma})}_N}$, then the natural map $\\mathcal{F}\\rightarrow s_{N,*}s_N^*\\mathcal{F}$ is an isomorphism. But this follows from the classical fact that the fibres of $s_N$ have cohomology equal to $\\mathbb{Q}$ in degree zero, and zero in other degrees.\n\\end{proof}\nWe now describe new cohomological Hall algebra operations on the underlying vector space\n\\[\n\\mathcal{T}^{\\Sp}_{Q,W}:=\\bigoplus_{\\gamma\\in\\mathbb{N}^{Q_0}}\\mathcal{T}^{\\Sp}_{Q,W,\\gamma}\n\\]\nwhere\n\\[\n\\mathcal{T}^{\\Sp}_{Q,W,\\gamma}:=\\left(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{Q,\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\right)^{\\SG_{\\gamma}}\\state{{-l(\\gamma)+\\chi(\\gamma,\\gamma)\/2}}.\n\\]\nFirstly, define \n\\[\n\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma}:=\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{Q,\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{-l(\\gamma)+\\chi(\\gamma,\\gamma)\/2}}.\n\\]\nAs always, we define the cohomological Hall algebra product as a composition of morphisms:\n\\begin{itemize}\n\\item\nDefine \n\\begin{align*}\n\\overline{\\delta}_T: &\\mathrm{H}_{c,T_{\\gamma}}(M_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+\\chi(\\gamma,\\gamma)\/2}}\\rightarrow \\\\&\\mathrm{H}_{c,T_{\\gamma}}(M_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*[\\eue(Q_0,\\gamma_1,\\gamma_2)^{-1}]\\state{{-l(\\gamma)+\\chi(\\gamma,\\gamma)\/2}}\n\\end{align*}\nto be division by $\\eue(Q_0,\\gamma_1,\\gamma_2)$.\n\\item\nDefine \n\\begin{align*}\n\\underline{\\overline{\\zeta}}_T:&\\mathrm{H}_{c,T_{\\gamma}}(M_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+\\chi(\\gamma,\\gamma)\/2}}\\rightarrow \\\\&\\mathrm{H}_{c,T_{\\gamma}}(M_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+\\chi(\\gamma,\\gamma)\/2}}\n\\end{align*}\nas the pushforward induced by the inclusion $M_{\\gamma_1,\\gamma_2}\\rightarrow M_{\\gamma}$.\n\n\\item\nDefine \n\\begin{align*}\n\\overline{\\alpha}_T:&\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1}\\times M^{\\Sp}_{\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}})^*\\state{{-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+l_1(\\gamma_1,\\gamma_2)+\\chi(\\gamma,\\gamma)\/2}}\\rightarrow\\\\& \\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+\\chi(\\gamma,\\gamma)\/2}}\n\\end{align*}\nas the pullback induced by the affine fibration $M_{\\gamma_1,\\gamma_2}\\rightarrow M_{\\gamma_1}\\times M_{\\gamma_2}$.\n\\end{itemize}\n\\begin{remark}\nThere is no $\\overline{\\beta}_T$ in the above list. This is because the $\\beta$ map (\\ref{betaMap}) is given by the pullback map induced by the passage from equivariant cohomology with respect to the $G_{\\gamma_1,\\gamma_2}$-action to equivariant cohomology with respect to the $G_{\\gamma_1}\\times G_{\\gamma_2}$-action. When we work with the torus-equivariant CoHA, there is no analogue of this move, since the isomorphism between $G_{\\gamma}$-equivariant dual compactly supported cohomology and $T_{\\gamma}$-equivariant dual compactly supported cohomology already involves a pullback map.\n\\end{remark}\nAn easy calculation shows that\n\\begin{align*}\n-l(\\gamma)+l_0(\\gamma_2,\\gamma_1)+l_1(\\gamma_2,\\gamma_1)+\\chi(\\gamma,\\gamma)\/2=&-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma_1,\\gamma_1)\/2+\\chi(\\gamma_2,\\gamma_2)\/2-\\\\&-\\chi(\\gamma_2,\\gamma_1)\/2+\\chi(\\gamma_1,\\gamma_2)\/2\n\\end{align*}\nand so the domain of $\\overline{\\alpha}_T$ is $\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes^{\\tw}_+\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_2}$, after applying the Thom--Sebastiani isomorphism.\nWe define \n\\begin{equation}\n\\label{mBarDef}\n\\overline{m}:\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_+^{\\tw}\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_2}\\rightarrow\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma}[\\eue(Q_0,\\gamma_1,\\gamma_2)^{-1}]\n\\end{equation}\nby $\\overline{m}= \\overline{\\delta}_T\\underline{\\overline{\\zeta}}_T\\overline{\\alpha}_T\\overline{\\TS}_T$. \n\\begin{example}\nConsider the example in which $W=0$, as in Section \\ref{NoPot}. Then \n\\[\n\\overline{\\mathcal{T}}_{\\gamma}=\\bigotimes_{i\\in Q_0} \\mathbb{Q}[x_{i,1},\\ldots,x_{i,\\gamma(i)}]\n\\]\nand the map (\\ref{mBarDef}) is given by\n\\begin{align}\n\\label{unSymmProd}\n&\\overline{m}(f_1,f_2)(x_{1,1},\\ldots,x_{1,\\gamma_1(1)+\\gamma_2(1)},\\ldots,x_{n,1},\\ldots,x_{n,\\gamma_1(n)+\\gamma_2(n)})=\\\\ \\nonumber&f_1(x_{1,1},\\ldots,x_{1,\\gamma_1(1)},\\ldots,x_{n,1},\\ldots,x_{n,\\gamma_1(n)})\\cdot\\\\&f_2(x_{1,\\gamma_1(1)+1},\\ldots,x_{1,\\gamma_1(1)+\\gamma_2(1)},\\ldots,x_{n,\\gamma_1(n)+1},\\ldots,x_{n,\\gamma_1(n)+\\gamma_2(n)})\\cdot\\\\ \\nonumber&\\prod_{i,j\\in Q_0}\\prod_{\\alpha=1}^{\\gamma_1(i)}\\nolimits\\prod_{\\beta=\\gamma_1(j)+1}^{\\gamma_1(j)+\\gamma_2(j)}\\nolimits(x_{j,\\beta}-x_{i,\\alpha})^{-b_{ij}}.\n\\end{align}\nNote that (\\ref{unSymmProd}) is obtained from (\\ref{expform}) by restricting to $\\SG_{\\gamma_1}$-invariant $f_1$ and $\\SG_{\\gamma_2}$-invariant $f_2$, and summing over shuffles in order to get a $\\SG_{\\gamma}$-invariant $m(f_1,f_2)$.\n\\end{example}\nReturning to the general case, the space $\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma}$ carries a $\\SG_{\\gamma}$-action, and by definition we have $\\mathcal{T}^{\\Sp}_{Q,W,\\gamma}:=\\overline{\\mathcal{T}}_{Q,W,\\gamma}^{\\Sp,\\SG_{\\gamma}}$. Each of the maps $\\overline{\\delta}_T,\\underline{\\overline{\\zeta}}_T,\\overline{\\alpha}_T,\\overline{\\TS}_T$ are $\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}$-equivariant, and restricting to invariant parts we define \n\\begin{align*}\n\\TS_T:=&\\overline{\\TS}_T^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}},\\\\\n\\alpha_T:=&\\overline{\\alpha}_T^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}},\\\\\n\\underline{\\zeta}_T:=&\\underline{\\overline{\\zeta}}_T^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}},\\\\\n\\delta_T:=&|\\mathcal{P}(\\gamma_1,\\gamma_2)|^{-1}\\left(\\sum_{\\pi\\in \\mathcal{P}(\\gamma_1,\\gamma_2)}\\pi\\right)\\overline{\\delta}_T^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}.\n\\end{align*}\nComposing these maps we build a map\n\\[\nm_T:=\\delta_T\\underline{\\zeta}_T\\alpha_T\\TS_T:\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_1}\\otimes\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_2}\\rightarrow (\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})_L^*)^{\\SG_{\\gamma}}\n\\]\nwhere the subscript $L$ means we formally invert $\\pi_*\\eue(Q_0,\\gamma_1,\\gamma_2)$ for every $\\pi\\in\\mathcal{P}(\\gamma_1,\\gamma_2)$.\n\\begin{proposition}\n\\label{TtoH}\nLet $\\gamma=\\gamma_1+\\gamma_2$. Then the following diagram commutes:\n\\begin{equation}\n\\label{TtoHd}\n\\xymatrix@C=10pt{\n(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})_L^*)^{\\SG_{\\gamma}}\\state{{\\spadesuit}}&\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{\\chi(\\gamma,\\gamma)\/2}\\ar[l]_-{\\xi_1}\n\\\\\n(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\\state{\\spadesuit+l_0(\\gamma_2,\\gamma_1)}\\ar[u]^{\\delta_T}&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2}}\\ar[u]^{\\delta}\\ar[l]_-{\\xi_2}\n\\\\\n\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*)^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\\state{\\diamondsuit+\\heartsuit-l_1(\\gamma_2,\\gamma_1)}\\ar[u]^{\\underline{\\zeta}_T}&\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2}}\\ar[u]^{\\underline{\\zeta}}\\ar[l]_-{\\xi_3}\n\\\\\n(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1}\\times M^{\\Sp}_{\\gamma_2},\\phi)^*)^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\\state{\\diamondsuit+\\heartsuit}\\ar[u]^{\\alpha_T}&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1}^{\\Sp}\\times M_{\\gamma_2}^{\\Sp},\\phi)\\state{{\\heartsuit}}^*\\ar[u]^{\\beta^{-1}\\alpha}\\ar[l]_-{\\xi_4}\n}\n\\end{equation}\n\n\nwhere in the last line $\\phi=\\phi_{\\tr(W)_{\\gamma_1}\\boxplus \\tr(W)_{\\gamma_2}}$, and all of the $\\xi_t$ for $t\\geq 2$ are isomorphisms, and the shifts are defined by\n\\begin{align*}\n\\spadesuit=&\\chi(\\gamma,\\gamma)-l(\\gamma)\n\\\\\n\\diamondsuit=&-l(\\gamma_1)-l(\\gamma_2),\n\\\\\n\\heartsuit=&\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1),\n\\end{align*}\n\\end{proposition}\nNote that the following identity follows from the definitions:\n\\[\n\\diamondsuit+\\heartsuit-l_1(\\gamma_2,\\gamma_1)=\\spadesuit+l_0(\\gamma_2,\\gamma_1),\n\\]\nso that the shifts in the domain and the target of $\\underline{\\zeta}_T$ agree.\n\\begin{proof}\nAll of the $\\xi_t$ are defined as follows. Firstly, we consider, for $V$ an arbitrary $\\SG_{\\gamma}$-equivariant vector space, the restriction map\n\\[\n(V^*)^{\\SG_{\\gamma}}\\rightarrow (V^{\\SG_{\\gamma}})^*.\n\\]\nThis defines a natural isomorphism of contravariant functors, since we always work over a field of characteristic zero. So we may interchange the operations of taking invariants and taking vector duals in the left hand column of (\\ref{TtoHd}). Then, we take the duals of the maps of Proposition \\ref{invprop}.\n\nWe deal with the commutativity of the constituent squares one by one, working from top to bottom. \n\n\\removelastskip\\vskip.5\\baselineskip\\par\\noindent{\\bf (Square 1)} For $V$ a $\\SG_{\\gamma}$-equivariant vector space, the following diagram commutes\n\\[\n\\xymatrix{\n(V^*)^{\\SG_{\\gamma}}\\ar[r]^{\\cong}&(V^{\\SG_{\\gamma}})^*\n\\\\\n(V^*)^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\\ar[u]^{|\\mathcal{P}(\\gamma_1,\\gamma_2)|^{-1}\\left(\\cdot \\sum_{\\pi\\in\\mathcal{P}(\\gamma_1,\\gamma_2)} \\pi\\right)}\\ar[r]^{\\cong}&(V^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}})^*,\\ar[u]^{i^*}\n}\n\\]\nwhere $i:V^{\\SG_{\\gamma}}\\rightarrow V^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}$ is the inclusion. It follows that we have a commutative diagram\n\\[\n\\xymatrix{\n(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma}}\\ar[r]^{\\cong}&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma}})^*\n\\\\\n(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\\ar[r]^{\\cong}\\ar[u]^{\\delta^+_T}\n&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma_1}\\times \\SG_{\\gamma_2}})^*\\ar[u]^{\\delta'_T}\n}\n\\]\nwhere $\\delta'_T$ is given by the dual of the inclusion\n\\[\n\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma}}\\rightarrow \\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}},\n\\]\ngiven by pushforward (in dual compactly supported cohomology) along the maps \n\\[\n\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N\\xrightarrow{b} \\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma}))}_N\n\\]\nand $\\delta_T^+=|\\mathcal{P}(\\gamma_1,\\gamma_2)|^{-1}\\left(\\sum_{\\pi\\in \\mathcal{P}(\\gamma_1,\\gamma_2)}\\pi\\right)$. Consider the following diagram\n\\[\n\\xymatrix{\nX_N\\ar@\/^1pc\/[drr]^{d'}\\ar@\/_2pc\/[rdd]_{c'}\\\\\n&\\ar[ul]^-i\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N\\ar[r]^-a\\ar[d]^b&\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma_1,\\gamma_2})}_N\\ar[d]^c\\\\\n&\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma}))}_N\\ar[r]^d&\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma})}_N\n}\n\\]\nwhere the perimeter of the diagram is a Cartesian square, i.e. \n\\[\nX_N:=\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma}))}_N\\times_{\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma})}_N}\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma_1,\\gamma_2})}_N.\n\\]\nThen by the proof of Proposition \\ref{invprop} the maps \n\\begin{align*}\n\\mathbb{Q}_{\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma_1,\\gamma_2})}_N}\\rightarrow &a_*\\mathbb{Q}_{\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N}\\\\\n\\mathbb{Q}_{\\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma_1,\\gamma_2})}_N}\\rightarrow &d'_*\\mathbb{Q}_{X_N}\n\\end{align*}\nare isomorphisms, and so in turn the map\n\\[\nd'_*\\mathbb{Q}_{X_N}\\rightarrow a_*\\mathbb{Q}_{\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N}\n\\]\nis an isomorphism. On the other hand, the map $i$ is a closed embedding, and the Euler characteristic of the normal bundle is $\\eue(Q_0,\\gamma_2,\\gamma_1)$: if we pull back along the inclusion of a point $x\\hookrightarrow \\overline{(M^{\\Sp}_{\\gamma},G_{\\gamma})}_N$ we obtain the following diagram\n\\[\n\\xymatrix{\nP(\\gamma)\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1,\\gamma)\\ar[drr]\\ar[ddr]\\\\\n& \\ar[ul]^-{i_x}P(\\gamma_1,\\gamma)\\ar[d]\\ar[r]^{a_x} &\\mathop{\\rm Gr}\\nolimits(\\gamma_1,\\gamma)\\ar[d]\\\\\n&P(\\gamma)\\ar[r]&x\n}\n\\]\nHere $\\mathop{\\rm Gr}\\nolimits(\\gamma_1,\\gamma)=\\prod_{i\\in Q_0} \\mathop{\\rm Gr}\\nolimits(\\mathbb{C}^{\\gamma_1},\\mathbb{C}^{\\gamma})$, and $P(\\gamma)=\\prod_{i\\in Q_0} P(\\gamma(i))$, where $P(n)$ is the space of $n$-tuples of unordered linearly independent lines in $\\mathbb{C}^n$, and $P(\\gamma_1,\\gamma)=\\prod_{i\\in Q_0}P(\\gamma_1(i),\\gamma(i))$, where $P(n',n)$ is the space of pairs $(T',T)$ where $T\\in P(n)$ and $T'\\subset T$ has order $n'$. The map $a_x$ is given by taking the span of the $T'$. So the inclusion $i_x$ is the inclusion of the space of pairs $(\\{T_i\\}_{i\\in Q_0},\\{V_i\\}_{i\\in Q_0})\\in P(\\gamma)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma_1,\\gamma)$ such that for each $i\\in Q_0$ the subspace $V_i\\subset\\mathbb{C}^{\\gamma(i)}$ is the space spanned by the first $\\gamma'_1(i)$ elements of $T_i$. The claim regarding the Euler characteristic of the normal bundle is then clear.\n\nBy Proposition \\ref{maneq} the following diagram commutes\n\\[\n\\xymatrix{\n\\mathrm{H}_c(X_N,\\phi_{\\tr(W)_N})^*\\ar[r]^-{i^*}\\ar[d]^{i^*}&\\mathrm{H}_c(\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N,\\phi_{\\tr(W)_N})^*\\ar[d]^{\\id}\\\\\n\\mathrm{H}_c(\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N,\\phi_{\\tr(W)_N})^*\\ar[r]^-{\\id}\\ar[d]^{i_*}&\\mathrm{H}_c(\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N,\\phi_{\\tr(W)_N})^*\\ar[d]^{\\cdot \\eue(Q_0,\\gamma_1,\\gamma_2)}\\\\\n\\mathrm{H}_c(X_N,\\phi_{\\tr(W)_N})^*\\ar[r]^-{i^*}&\\mathrm{H}_c(\\overline{(M^{\\Sp}_{\\gamma},N(T_{\\gamma_1})\\times N(T_{\\gamma_2}))}_N,\\phi_{\\tr(W)_N})^*.\n}\n\\]\nThe top square trivially commutes, while commutativity of the bottom square is the content of Proposition \\ref{maneq}. The horizontal maps are isomorphisms as in Proposition \\ref{invprop}, so we deduce that the composition of the leftmost vertical arrows is also given by multiplication by the Euler class $\\eue(Q_0,\\gamma_1,\\gamma_2)$. Putting everything together, we deduce that\n\\begin{align*}\nb_*a^*=&c'_*i_*i^*d'^*\\\\\n=&c'_*d'^*\\circ (\\cdot\\eue(Q_0,\\gamma_1,\\gamma_2))\\\\\n=&d^*c_*\\circ (\\cdot\\eue(Q_0,\\gamma_1,\\gamma_2))\n\\end{align*}\nwhere the equality $c'_*d'^*=d^*c_*$ is as in Proposition \\ref{mixingprop}, so that we have the equality\n\\[\nb_*a^*=d^*c_*\\circ(\\cdot \\eue(Q_0,\\gamma_1,\\gamma_2))\n\\]\nand so\n\\[\n\\delta\\epsilon_1\\circ (\\cdot \\eue(Q_0,\\gamma_1,\\gamma_2))=\\epsilon_2\\delta_T^+.\n\\]\nIt follows that after localising we have the equality\n\\[\n\\delta\\epsilon_1=\\epsilon_2\\delta_T.\n\\]\nas required.\n\n\n\n\\removelastskip\\vskip.5\\baselineskip\\par\\noindent{\\bf (Square 2)} The following diagram of spaces is a commutative Cartesian diagram, in which the vertical maps are closed inclusions and the horizontal maps are smooth projections:\n\\[\n\\xymatrix{\n\\overline{(M^{\\Sp}_{\\gamma},\\No(T_{\\gamma}))}_N\\ar[r]&\\overline{(M^{\\Sp}_{\\gamma}, G_{\\gamma_1,\\gamma_2})}_N\\\\\n\\overline{(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\No(T_{\\gamma}))}_N\\ar[r]\\ar[u]&\\overline{(M^{\\Sp}_{\\gamma_1,\\gamma_2}, G_{\\gamma_1,\\gamma_2})}_N\\ar[u]\n}\n\\]\nand $\\zeta_T$, $\\zeta$ are obtained by pushforward along the vertical maps, while $\\xi_2$ and $\\xi_3$ are obtained by pullback along the horizontal arrows. Commutativity then follows by Proposition \\ref{mixingprop}.\n\n\n\n\n\\removelastskip\\vskip.5\\baselineskip\\par\\noindent{\\bf (Square 3)} The proof of the commutativity of the bottom square is as in the proof of the commutativity of the second square, using Proposition \\ref{mixingprop} in the affine fibration case.\n\\end{proof}\nSince the map $\\epsilon_1\\delta$ lands in $\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{Q,\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*$ we deduce the following corollaries.\n\\begin{corollary}\n\\label{Tfactor}\nThe map\n\\[\nm_T:\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_+^{\\tw}\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_2}\\rightarrow(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})_L^*)^{\\SG_{\\gamma}}\n\\]\nfactors through $\\mathcal{T}^{\\Sp}_{Q,W,\\gamma}$, and induces an associative multiplication on $\\mathcal{T}^{\\Sp}_{Q,W}$, which we will also denote $m_T$. \n\\end{corollary}\n\\begin{corollary}\n\\label{PsiDef}\nThere is an isomorphism of algebras $\\Psi:(\\mathcal{T}^{\\Sp}_{Q,W},m_T)\\cong(\\mathcal{H}^{\\Sp}_{Q,W},m)$. Equivalently, the composition\n\\[\n\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_{+}^{\\tw}\\mathcal{T}_{Q,W,\\gamma_2}^{\\Sp}\\xrightarrow{\\cong}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\boxtimes_{+}^{\\tw}\\mathcal{H}_{Q,W,\\gamma_2}^{\\Sp}\\xrightarrow{m}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}\\rightarrow (\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})_L^*)^{\\SG_{\\gamma}}\n\\]\nis equal to the map $m_T$.\n\\end{corollary}\n\n\\section{The comultiplication on $\\mathcal{H}^{\\Sp}_{Q,W}$}\n\\label{comult_section}\n\n\\subsection{$Q$-localised bialgebras}\n\\label{QLocBi}\nA $\\mathbb{Q}$-linear tensor category $\\mathcal{C}$ is a symmetric monoidal $\\mathbb{Q}$-linear category with a monoidal unit $\\mathbf{1}_{\\mathcal{C}}$, for which the monoidal product is exact in both arguments. Let $\\mathcal{C}$ be an Abelian $\\mathbb{Q}$-linear tensor category, closed under small products and coproducts, with an exact faithful $\\mathbb{Q}$-linear tensor functor $\\fib:\\mathcal{C}\\rightarrow\\Vect$ to the tensor category of $\\mathbb{Q}$-vector spaces.\n\\smallbreak\nWe define $A_{\\gamma}:=\\mathbb{Q}[x_{1,1},\\ldots,x_{1,\\gamma(1)},\\ldots,x_{n,1},\\ldots,x_{n,\\gamma(n)}]^{\\SG_{\\gamma}}$. Throughout we consider $A_{\\gamma}$ as an object of $\\mathcal{C}$ in the obvious way: we first form $\\overline{A}_{\\gamma}$, the free unital commutative algebra in $\\mathcal{C}$ generated by a copy of $\\mathbb{Q}\\state{{-1}}$ for each generator $x_{h,j}$ in $\\mathbb{Q}[x_{1,1},\\ldots,x_{1,\\gamma(1)},\\ldots,x_{n,1},\\ldots,x_{n,\\gamma(n)}]$, and then define $A_{\\gamma}$ to be the $\\SG_{\\gamma}$ invariant part of $\\overline{A}_{\\gamma}$, which we can define, abstractly, as a kernel in each cohomological degree. For $\\gamma_1,\\ldots,\\gamma_r\\in\\mathbb{N}^{Q_0}$ an $r$-tuple of dimension vectors we define\n\\[\nA_{\\gamma_1,\\ldots,\\gamma_r}=\\bigotimes_{\\tau=1,\\ldots,r}\\mathbb{Q}[x^{(\\tau)}_{1,1},\\ldots,x^{(\\tau)}_{1,\\gamma_{\\tau}(1)},\\ldots,x^{(\\tau)}_{n,1},\\ldots,x^{(\\tau)}_{n,\\gamma_{\\tau}(n)}]^{\\SG_{\\gamma_{\\tau}}}.\n\\]\nWe define the localised monoidal product as follows:\n\\[\nB'\\tilde{\\boxtimes}^{\\tw}_+B''=\\bigoplus_{\\gamma_1,\\gamma_2\\in\\mathbb{N}^{Q_0}}B'_{\\gamma_1}\\boxtimes_+^{\\tw} B''_{\\gamma_2}\\otimes_{A_{\\gamma_1,\\gamma_2}} A_{\\gamma_1,\\gamma_2}\\left[\\prod_{i,j\\in Q_0}\\prod_{\\substack{s'=1,\\ldots,\\gamma_1(j)\\\\s''=1,\\ldots,\\gamma_2(i)}}\\left(x^{(1)}_{j,s'}-x^{(2)}_{i,s''}\\right)^{-1}\\right]\n\\]\nwhere we consider the localised algebra $A_{\\gamma_1,\\gamma_2}[\\prod_{i,j\\in Q_0}\\prod_{\\substack{s'=1,\\ldots,\\gamma_1(j)\\\\s''=1,\\ldots,\\gamma_2(i)}}(x^{(1)}_{j,s'}-x^{(2)}_{i,s''})^{-1}]$ as a cohomologically graded object of $\\mathcal{C}$ in the same way as $A_{\\gamma_1,\\gamma_2}$. More generally, if $\\gamma_1,\\ldots,\\gamma_r\\in\\mathbb{N}^{Q_0}$ and \n\\[\nS\\subset \\{(\\tau,\\mu)\\in\\{1,\\ldots,r\\}^{\\times 2}|\\tau\\neq \\mu\\}\n\\]\nthen we define\n\\begin{align*}\n&[B_{1}\\boxtimes^{\\tw}_+\\ldots\\boxtimes^{\\tw}_+ B_{r}]_{S}:=\\\\&\\bigoplus_{\\gamma_1,\\ldots,\\gamma_r\\in\\mathbb{N}^{Q_0}} B_{\\gamma_1}\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw} B_{\\gamma_r}\\otimes_{A_{\\gamma_1,\\ldots,\\gamma_r}}A_{\\gamma_1,\\ldots,\\gamma_r}\\left[\\prod_{i,j\\in Q_0}\\prod_{(\\tau,\\mu)\\in S}\\prod_{\\substack{s'=1,\\ldots\\gamma_{\\tau}(j)\\\\s''=1,\\ldots,\\gamma_{\\mu}(i)}}(x^{(\\tau)}_{j,s'}-x^{(\\mu)}_{i,s''})^{-1}\\right].\n\\end{align*}\n\nRecall that the monoidal product $\\boxtimes_+^{\\tw}$ is not symmetric. In contrast, under the above assumption of $A_{\\gamma}$-actions on the $B_{\\gamma}$, there is a natural isomorphism \n\\begin{equation}\n\\label{newsw}\n\\tilde{\\sw}_{\\gamma_1,\\gamma_2}:B_{\\gamma_1}\\tilde{\\boxtimes}^{\\tw}_+ B_{\\gamma_2}\\rightarrow B_{\\gamma_2}\\tilde{\\boxtimes}^{\\tw}_+ B_{\\gamma_1}\n\\end{equation}\ngiven by the composition $\\cdot\\tilde{\\eue}_{\\gamma_1,\\gamma_2} \\circ\\sw_{\\boxtimes_+}$ where $\\cdot\\tilde{\\eue}_{\\gamma_1,\\gamma_2}$ is multiplication by\n\\begin{equation}\n\\label{corr_term}\n(-1)^{\\gamma_1\\cdot\\gamma_2}\\prod_{i,j\\in Q_0}\\limits\\left(\\prod_{m=1}^{\\gamma_1(j)}\\nolimits\\prod_{m'=1}^{\\gamma_2(i)}\\nolimits(x^{(1)}_{j,m}-x^{(2)}_{i,m'})^{-b_{ij}}(x^{(2)}_{j,m'}-x^{(1)}_{i,m})^{b_{ij}}\\right).\n\\end{equation}\n\\begin{remark}\nWe may express $\\tilde{\\eue}_{\\gamma_1,\\gamma_2}$ in terms of the operators $\\eue(Q_1,\\bullet,\\bullet)$ and $\\eue(Q_0,\\bullet,\\bullet)$ as follows:\n\\begin{equation}\n\\label{lwdd}\n\\tilde{\\eue}_{\\gamma',\\gamma''}=\\eue(Q_1,\\gamma_2,\\gamma_1)^{-1}\\eue(Q_0,\\gamma_2,\\gamma_1)\\eue(Q_1,\\gamma_1,\\gamma_2)\\eue(Q_0,\\gamma_1,\\gamma_2)^{-1}.\n\\end{equation}\nEquality between (\\ref{lwdd}) and (\\ref{corr_term}) follows from the equality \n\\[\n\\eue(Q_0,\\gamma_2,\\gamma_1)\\eue(Q_0,\\gamma_1,\\gamma_2)^{-1}=(-1)^{\\gamma_1\\cdot\\gamma_2}.\n\\]\n\\end{remark}\nTaking the direct sum of (\\ref{newsw}) over all pairs $\\gamma_1,\\gamma_2\\in\\mathbb{N}^{Q_0}$ we define an isomorphism \n\\begin{align*}\n\\tilde{\\sw}:=\\bigoplus_{\\gamma_1,\\gamma_2\\in\\mathbb{N}^{Q_0}}\\tilde{\\sw}_{\\gamma_1,\\gamma_2}:&B\\tilde{\\boxtimes_+^{\\tw}}B\\rightarrow B\\tilde{\\boxtimes_+^{\\tw}}B, \\end{align*}\nswapping pairs of dimension vectors, with $\\tilde{\\sw}^2=\\id$. More generally, for $\\gamma_1,\\ldots,\\gamma_r\\in\\mathbb{N}^{Q_0}$ and $(\\tau,\\mu)\\in S$, we define $\\tilde{\\sw}_{\\tau\\mu}:[B_{1}\\boxtimes^{\\tw}_+\\ldots\\boxtimes^{\\tw}_+ B_{r}]_{S}\\rightarrow [B_{1}\\boxtimes^{\\tw}_+\\ldots\\boxtimes^{\\tw}_+ B_{r}]_{(\\tau\\mu)_*S}$ by swapping the $\\tau$ and the $\\mu$ factors and multiplying by \n\\begin{equation}\n(-1)^{\\gamma_{\\tau}\\cdot\\gamma_{\\mu}}\\prod_{i,j\\in Q_0}\\limits\\left(\\prod_{m=1}^{\\gamma_{\\tau}(j)}\\nolimits\\prod_{m'=1}^{\\gamma_{\\mu}(i)}\\nolimits(x^{(\\tau)}_{j,m}-x^{(\\mu)}_{i,m'})^{b_{ij}}(x^{(\\mu)}_{j,m'}-x^{(\\tau)}_{i,m})^{-b_{ij}}\\right).\n\\end{equation}\n\\begin{definition}\nLet $(B,m)$ be an algebra in $\\mathcal{C}_Q$ with respect to the monoidal structure $\\boxtimes_+^{\\tw}$, such that each $\\mathbb{N}^{Q_0}$-graded piece $B_{\\gamma}$ carries an $A_{\\gamma}$-action. We say that $B$ is crosslinear if for all $i,j\\in Q_0$, the map\n\\[\nB_{\\gamma_1}\\boxtimes_+^{\\tw}B_{\\gamma_2}\\boxtimes_+^{\\tw}B_{\\gamma_3}\\boxtimes_+^{\\tw}B_{\\gamma_4}\\xrightarrow{m\\boxtimes_+^{\\tw}m}B_{\\gamma_1+\\gamma_2}\\boxtimes_+^{\\tw}B_{\\gamma_3+\\gamma_4}\n\\]\ncommutes with multiplication by \n\\begin{equation}\n\\label{easa}\n\\prod_{i,j\\in Q_0}\\prod_{\\substack{\\tau\\in\\{1,2\\}\\\\ \\mu\\in\\{3,4\\}}}\\prod_{m=1}^{\\gamma_{\\tau}(i)}\\prod_{m'=1}^{\\gamma_{\\mu}(j)}(x_{i,m}^{(\\tau)}-x_{j,m'}^{(\\mu)}).\n\\end{equation}\nHere we use the natural isomorphisms, for $T$ any ordered finite set,\n\\begin{align*}\n&\\bigotimes_{\\tau\\in T}\\mathbb{Q}[x^{(\\tau)}_{1,1},\\ldots,x^{(\\tau)}_{1,\\gamma_{\\tau}(1)},\\ldots,x^{(\\tau)}_{n,1},\\ldots,x^{(\\tau)}_{n,\\gamma_{\\tau}(n)}]\\cong \\\\& \\mathbb{Q}[x_{1,1},\\ldots,x_{1,\\sum_{\\tau\\in T}\\gamma_{\\tau}(1)},\\ldots,x_{n,1},\\ldots,x_{n,\\sum_{\\tau\\in T}\\gamma_{\\tau}(n)}]\n\\end{align*}\nto realise (\\ref{easa}) as an element of $A_{\\gamma_1+\\gamma_2,\\gamma_3+\\gamma_4}$.\\end{definition}\nIf $B$ is a crosslinear algebra with respect to the monoidal structure $\\boxtimes_+^{\\tw}$, then we define the product $\\tilde{m}^2$ on $B\\tilde{\\boxtimes}_+^{\\tw}B$ via the composition\n\\[\n\\xymatrix{\n\\left(B\\tilde{\\boxtimes}_+^{\\tw}B\\right)\\boxtimes_+^{\\tw}\\left(B\\tilde{\\boxtimes}_+^{\\tw}B\\right)\\ar[r]^-=&[B\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw} B]_{(1,2),(3,4)}\\ar[r] &[B\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw} B]_{(1,2),(3,2),(3,4),(1,4)}\\ar[dll]_{\\tilde{\\sw}_{(23)}}\\\\\n[B\\boxtimes_+^{\\tw}\\ldots\\boxtimes_+^{\\tw} B]_{(1,3),(2,3),(2,4),(1,4)}\\ar[r]_-{m\\tilde{\\boxtimes}_+^{\\tw}m}&B\\tilde{\\boxtimes}_+^{\\tw}B\n}\n\\]\nwhere $m\\tilde{\\boxtimes}_+^{\\tw}m$ is the unique extension of $m\\boxtimes_+^{\\tw}m$ to the localisation with respect to \n\\[\n\\prod_{i,j\\in Q_0}\\prod_{\\substack{\\tau\\in\\{1,2\\}\\\\ \\mu\\in\\{3,4\\}}}\\prod_{m=1}^{\\gamma_{\\tau}(i)}\\prod_{m'=1}^{\\gamma_{\\mu}(j)}(x_{i,m}^{(\\tau)}-x_{j,m'}^{(\\mu)})^{-1},\n\\]\nwhich exists by the assumption of crosslinearity for $m$.\n\\begin{definition}\n\\label{lbs}\nA $Q$-localised bialgebra in $\\mathcal{C}$ is the data of a unital algebra object $(B,m,\\nu:\\mathbf{1}\\rightarrow B)$ in $\\mathcal{C}_{Q}$ and an $A_{\\gamma}$-module structure for each $B_{\\gamma}$, such that $B$ is crosslinear. We require also the data of a morphism\n\\begin{equation}\n\\Delta:B\\rightarrow B\\tilde{\\boxtimes}^{\\tw}_+ B,\n\\end{equation}\nsuch that each map\n\\[\n\\Delta_{\\gamma_1,\\gamma_2}:B_{\\gamma_1+\\gamma_2}\\rightarrow B_{\\gamma_1}\\tilde{\\boxtimes}_+^{\\tw}B_{\\gamma_2}\n\\]\nis $A_{\\gamma}$-linear, and the diagram\n\\[\n\\xymatrix{\nB\\ar[d]^{\\Delta}\\ar[rr]^-{\\Delta}&& B\\tilde{\\boxtimes}_+^{\\tw}B\\ar[d]^{\\id\\tilde{\\boxtimes}_+^{\\tw}\\Delta}\n\\\\\nB\\tilde{\\boxtimes}_+^{\\tw}B\\ar[rr]^-{\\Delta\\tilde{\\boxtimes}_+^{\\tw}\\id}&&[B\\boxtimes_+^{\\tw}B\\boxtimes_+^{\\tw}B]_{\\{(1,2),(1,3),(2,3)\\}}\n}\n\\]\ndefined via this linearity commutes. Finally, we require that the following diagram commutes\n\\begin{equation}\n\\label{commdiagB}\n\\xymatrix{\nB\\boxtimes_+^{\\tw} B\\ar[d]^m\\ar[r]^-{\\Delta\\boxtimes_+^{\\tw}\\Delta}&B\\tilde{\\boxtimes}_+^{\\tw} B\\boxtimes_+^{\\tw} B\\tilde{\\boxtimes}_+^{\\tw} B\\ar[d]^{\\tilde{m}^2}\\\\\nB\\ar[r]^-{\\Delta}&B\\tilde{\\boxtimes}_+^{\\tw} B.\n}\n\\end{equation}\nWe require also that for all $S\\subset \\{(\\tau,\\mu)\\in\\{1,\\ldots,r\\}^{\\times 2}|\\tau\\neq \\mu\\}$ the localisation map\n\\begin{equation}\n\\label{reqinj}\nB^{\\boxtimes_+^{\\tw} r}\\rightarrow [B^{\\boxtimes_+^{\\tw} r}]_S\n\\end{equation}\nis injective.\n\\end{definition}\nBefore introducing the comultiplication on $\\mathcal{H}^{\\Sp}_{Q,W}$, we verify that the background assumptions in the definition of a $Q$-localised bialgebra on the algebra $\\mathcal{H}_{Q,W}^{\\Sp}$ apply. The injectivity of the map (\\ref{reqinj}) is just the statement of Proposition \\ref{ABprop}, so we will concentrate on the crosslinearity condition.\n \n\n\n\n\n\\begin{proposition}\n\\label{coCross}\nThe Cohomological Hall algebra $\\mathcal{H}^{\\Sp}_{Q,W}$ is crosslinear.\n\\end{proposition}\n\\begin{proof}\nThe class\n\\[\n\\bigstar=\\prod_{i,j\\in Q_0}\\prod_{\\substack{\\tau\\in\\{1,2\\}\\\\ \\mu\\in\\{3,4\\}}}\\prod_{m=1}^{\\gamma_{\\tau}(i)}\\prod_{m'=1}^{\\gamma_{\\mu}(j)}(x_{i,m}^{(\\tau)}-x_{j,m'}^{(\\mu)})\n\\]\nconsidered as an element of $\\mathrm{H}_{G_{\\gamma_1}}(\\mathop{\\rm pt})\\otimes\\ldots\\otimes\\mathrm{H}_{G_{\\gamma_4}}(\\mathop{\\rm pt})$ is equal to the pullback of the class $\\bigstar$, considered as an element of $\\mathrm{H}_{G_{\\gamma_1+\\gamma_2}}(\\mathop{\\rm pt})\\otimes\\mathrm{H}_{G_{\\gamma_3+\\gamma_4}}(\\mathop{\\rm pt})$ under the natural map\n\\[\n\\mathrm{H}_{\\gamma_1+\\gamma_2}(\\mathop{\\rm pt})\\otimes\\mathrm{H}_{\\gamma_3+\\gamma_4}(\\mathop{\\rm pt})\\rightarrow\\mathrm{H}_{G_{\\gamma_1}}(\\mathop{\\rm pt})\\otimes\\ldots\\otimes\\mathrm{H}_{G_{\\gamma_4}}(\\mathop{\\rm pt}).\n\\]\nThe result then follows from Proposition \\ref{maneq}.\n\\end{proof}\n\n\n\n\n\n\\subsection{Comultiplication operation on $\\mathcal{H}^{\\Sp}_{Q,W}$}\n\nFor each decomposition $\\gamma=\\gamma_1+\\gamma_2$ we define a map \n\\begin{equation}\n\\Delta_{\\gamma_1,\\gamma_2}:\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}\\rightarrow \\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\tilde{\\boxtimes_+^{\\tw}} \\mathcal{H}^{\\Sp}_{Q,W,\\gamma_2}\n\\end{equation}\nthat we will show defines the structure of a $Q$-localised comultiplication for $\\mathcal{H}_{Q,W}^{\\Sp}$.\n\n\n\nTo begin, we work with the torus-equivariant critical cohomological Hall algebra $\\mathcal{T}^{\\Sp}_{Q,W}\\cong\\mathcal{H}^{\\Sp}_{Q,W}$. We define\n\\begin{align*}\n\\overline{\\Delta}_{T,\\gamma_1,\\gamma_2}:&\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{Q,\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{-l(\\gamma)+\\chi(\\gamma,\\gamma)}}\\rightarrow\\\\& \\mathrm{H}_{c,T_{\\gamma_1}}(M^{\\Sp}_{Q,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_1}})^*\\tilde{\\boxtimes}_+^{\\tw}\\mathrm{H}_{c,T_{\\gamma_2}}(M^{\\Sp}_{Q,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_2}})^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma_1,\\gamma_1)\/2+\\chi(\\gamma_2,\\gamma_2)\/2}}\n\\end{align*}\nas the composition of the following maps:\n\\begin{itemize}\n\\item\nDefine \n\\begin{align*}\n\\overleftarrow{\\alpha}_T:&\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}} \\rightarrow \\\\&\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_1}\\times M^{\\Sp}_{\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}}\\right)^*[\\eue(Q_1,\\gamma_1,\\gamma_2)^{-1}]\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\n\\end{align*}\nas the pushforward associated to the affine fibration\\footnote{Note that this is not the same affine fibration we used to define $\\alpha$.} $M_{\\gamma_2,\\gamma_1}\\xrightarrow{\\pi} M_{\\gamma_1}\\times M_{\\gamma_2}$. Note that\n\\begin{align*}\n&-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma_1,\\gamma_1)\/2+\\chi(\\gamma_2,\\gamma_2)\/2-\\chi(\\gamma_2,\\gamma_1)\/2+\\chi(\\gamma_1,\\gamma_2)\/2=\\\\&-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)\n\\end{align*}\nso that the target of $\\overleftarrow{\\alpha}_T$ is the same as the target of $\\overline{\\Delta}_{T,\\gamma_1,\\gamma_2}$ after applying the inverse of the Thom--Sebastiani isomorphism. Here we are using that $\\eue(Q_1,\\gamma_1,\\gamma_2)$ is the Euler class of $\\pi$, which has an obvious $T_{\\gamma}$-equivariant section given by the inclusion of block diagonal matrices into block lower triangular matrices.\n\\item\nDefine \n\\begin{align*}\n&\\overleftarrow{\\beta^{-1}}_T:\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)-l_0(\\gamma_2,\\gamma_1)}}\\rightarrow \\\\&\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)}}\n\\end{align*}\nto be multiplication by \n\\[\n\\prod_{i\\in Q_0}\\prod_{m=1}^{\\gamma_1(i)}\\nolimits\\prod_{m'=1}^{\\gamma_2(i)}\\nolimits(x_{i,m}-x_{i,m'+\\gamma_1(i)})=\\eue(Q_0,\\gamma_2,\\gamma_1),\n\\]\nwhere here we have made the natural identification\n\\[\nA_{\\gamma_1,\\gamma_2}=\\left(\\bigotimes_{i\\in Q_0} \\mathbb{Q}[x_{i,1},\\ldots,x_{i,\\gamma_1(i)+\\gamma_2(i)}]\\right)^{{\\rm Sym}_{\\gamma_1}\\times{\\rm Sym}_{\\gamma_2}}.\n\\]\n\n\\item\nDefine \n\\begin{align*}\n\\overleftarrow{\\underline{\\zeta}}_T: &\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}}\\right)^*\\state{{-l(\\gamma)+\\chi(\\gamma,\\gamma)\/2}}\\rightarrow \\\\&\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)-l_0(\\gamma_2,\\gamma_1)}}\n\\end{align*}\nas the pullback induced by the inclusion $M^{\\Sp}_{\\gamma_2,\\gamma_1}\\rightarrow M^{\\Sp}_{\\gamma}$.\n\\item\nDefine \n\\begin{align*}\n\\overleftarrow{\\delta}_T: &\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)+l_0(\\gamma_2,\\gamma_1)}}\\rightarrow\\\\ &\\mathrm{H}_{c,T_{\\gamma}}\\left(M^{\\Sp}_{\\gamma_2,\\gamma_1},\\phi_{\\tr(W)_{\\gamma_2,\\gamma_1}}\\right)^*\\state{{-l(\\gamma_1)-l(\\gamma_2)+\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_2,\\gamma_1)+l_0(\\gamma_2,\\gamma_1)}}\n\\end{align*}\nto be the identity map.\n\\end{itemize}\n\n\\begin{definition}\nThe map \n\\[\n\\overline{\\Delta}_{T,\\gamma_1,\\gamma_2}:\\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma}\\rightarrow \\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_1}\\tilde{\\boxtimes}_+^{\\tw} \\overline{\\mathcal{T}}^{\\Sp}_{Q,W,\\gamma_2}\n\\]\nis defined by $\\overline{\\Delta}_{T,\\gamma_1,\\gamma_2}=\\overline{\\TS}^{-1}\\overleftarrow{\\alpha}_T\\overleftarrow{\\beta^{-1}}_T\\overleftarrow{\\underline{\\zeta}}_T\\overleftarrow{\\delta}_T\n$\nand \n\\[\n\\Delta_{T,\\gamma_1,\\gamma_2}:\\mathcal{T}^{\\Sp}_{Q,W}\\rightarrow \\bigoplus_{\\gamma_1,\\gamma_2\\in\\mathbb{Z}^{Q_0}}\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_1}\\tilde{\\boxtimes}_+^{\\tw} \\mathcal{T}^{\\Sp}_{Q,W,\\gamma_2}\n\\]\nis defined by restricting to the $\\SG_{\\gamma}$ invariant part of the domain.\nThe map\n\\[\n\\Delta_T:\\mathcal{T}^{\\Sp}_{Q,W}\\rightarrow \\bigoplus_{\\gamma_1,\\gamma_2\\in\\mathbb{Z}^{Q_0}}\\mathcal{T}^{\\Sp}_{Q,W,\\gamma_1}\\tilde{\\boxtimes}_+^{\\tw} \\mathcal{T}^{\\Sp}_{Q,W,\\gamma_2}\n\\]\nis defined to be the sum\n\\begin{equation}\n\\label{Deltadef}\n\\Delta_T=\\sum_{\\gamma_1+\\gamma_2=\\gamma}\\Delta_{T,\\gamma_1,\\gamma_2}.\n\\end{equation}\n\\end{definition}\n\\begin{definition}\nWe define the comultiplication \n\\[\n\\Delta:\\mathcal{H}_{Q,W}^{\\Sp}\\rightarrow \\mathcal{H}_{Q,W}^{\\Sp}\\tilde{\\boxtimes}^{\\tw}_{+}\\mathcal{H}_{Q,W}^{\\Sp}\n\\]\nvia the formula\n\\[\n\\Delta:=(\\Psi\\tilde{\\boxtimes}^{\\tw}_{+}\\Psi)\\circ \\Delta_T\\circ \\Psi^{-1},\n\\]\nwhere $\\Psi$ is the isomorphism of Corollary \\ref{PsiDef}.\n\\end{definition}\nSince each of the maps defining $\\overline{\\Delta}_T$ is $\\mathrm{H}_{T_{\\gamma}}(\\mathop{\\rm pt})$-linear, and hence also $\\mathrm{H}_{T_{\\gamma}}(\\mathop{\\rm pt})^{\\SG_{\\gamma}}$-linear, we deduce that $\\Delta$ is $A_{\\gamma}:=\\mathrm{H}_{G_{\\gamma}}(\\mathop{\\rm pt})$-linear. The following is proved in just the same way as the associativity of $m$, see \\cite{COHA}.\n\\begin{proposition}\nThe following diagram commutes\n\\[\n\\xymatrix{\n\\mathcal{H}_{Q,W}^{\\Sp}\\ar[rr]^-{\\Delta}\\ar[d]_{\\Delta}&&\\mathcal{H}_{Q,W}^{\\Sp}\\tilde{\\boxtimes}_+^{\\tw}\\mathcal{H}_{Q,W}^{\\Sp}\\ar[d]^-{\\id\\tilde{\\boxtimes}^{\\tw}_+\\Delta}\n\\\\\n\\mathcal{H}_{Q,W}^{\\Sp}\\tilde{\\boxtimes}_+^{\\tw}\\mathcal{H}_{Q,W}^{\\Sp}\n\\ar[rr]^-{\\Delta\\tilde{\\boxtimes}^{\\tw}_+\\id}\n&&\n[\\mathcal{H}^{\\Sp}_{Q,W}\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W}\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W}]_{\\{(1,2),(1,3),(2,3)\\}}.\n}\n\\]\n\\end{proposition}\n\n\\subsection{Another description of the coproduct}\nWe now describe directly the product \n\\[\n\\Delta:\\mathcal{H}^{\\Sp}_{Q,W}\\rightarrow \\mathcal{H}^{\\Sp}_{Q,W}\\tilde{\\boxtimes_+^{\\tw}}\\mathcal{H}_{Q,W}^{\\Sp}\n\\]\nwithout reference to $\\mathcal{T}^{\\Sp}_{Q,W}$. The morphisms $\\alpha,\\beta,\\delta,\\underline{\\zeta}$ are as defined in Section \\ref{MOsec}. By replacing pushforwards by pullbacks, and vice versa, we arrive at morphisms\n\\begin{align*}\n\\label{pfdef}\n\\overleftarrow{\\delta}: &\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)-l_1(\\gamma_1,\\gamma_2)+l_0(\\gamma_1,\\gamma_2)}\\rightarrow\\\\& \\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)-l_1(\\gamma_1,\\gamma_2)}\n\\\\\\\\\n\\overleftarrow{\\underline{\\zeta}}: &\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)-l_1(\\gamma_1,\\gamma_2)}\\rightarrow\\\\& \\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)}\n\\\\\\\\\n\\overleftarrow{\\alpha}: &\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1,\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)}\\rightarrow \\\\&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M_{\\gamma_1}^{\\Sp}\\times M_{\\gamma_2}^{\\Sp},\\phi_{\\tr(W)_{\\gamma_1}\\boxplus\\tr(W)_{\\gamma_2}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2)}}.\n\\end{align*}\n\\begin{proposition}\n\\label{UnT}\nThe following diagram commutes:\n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2}}\\ar[d]^{\\overleftarrow{\\delta}}\\ar[r]&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma}}\\state{{\\spadesuit}}\\ar[d]^{\\overleftarrow{\\delta}_T}\n\\\\\n\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\chi(\\gamma,\\gamma)\/2-l_0(\\gamma_1,\\gamma_2)}}\\ar[d]^{\\overleftarrow{\\underline{\\zeta}}}\\ar[r]&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma}}\\state{{\\spadesuit}}\\ar[d]^{\\overleftarrow{\\underline{\\zeta}}_T}\n\\\\\n\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma}})^*\\state{{\\heartsuit}}\\ar[d]^{\\beta\\cdot \\eue(Q_0,\\gamma_2,\\gamma_1)}\\ar[r]&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma}})^*)^{\\SG_{\\gamma}}\\state{{\\heartsuit+\\diamondsuit-l_1(\\gamma_1,\\gamma_2)}}\\ar[d]^{\\overleftarrow{\\beta^{-1}}_T}\n\\\\\n\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*\\state{{\\heartsuit}}\\ar[d]^{\\overleftarrow{\\alpha}}\\ar[r]&(\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\gamma_1,\\gamma_2},\\phi_{\\tr(W)_{\\gamma_1,\\gamma_2}})^*)^{\\SG_{\\gamma}}\\state{{\\heartsuit+\\diamondsuit}}\\ar[d]^{\\overleftarrow{\\alpha}_T}\n\\\\\n\\mathcal{H}^{\\Sp}_{\\gamma_1}\\tilde{\\boxtimes_+^{\\sw}}\\mathcal{H}^{\\Sp}_{\\gamma_2}\\ar[r]&\\mathcal{T}^{\\Sp}_{\\gamma_1}\\tilde{\\boxtimes_+^{\\sw}}\\mathcal{T}^{\\Sp}_{\\gamma_2}\n}\n\\]\nwhere \n\\[\n\\diamondsuit=-l(\\gamma_1)-l(\\gamma_2),\n\\]\n\\[\n\\heartsuit=\\chi(\\gamma,\\gamma)\/2-\\chi(\\gamma_1,\\gamma_2),\n\\]\n\\begin{align*}\n\\spadesuit=&\\heartsuit+\\diamondsuit-l_0(\\gamma_1,\\gamma_2)-l_1(\\gamma_1,\\gamma_2)\\\\\n=&\\chi(\\gamma,\\gamma)\/2-l(\\gamma),\n\\end{align*}\nand the horizontal maps are the isomorphisms of Proposition \\ref{TtoH}.\n\\end{proposition}\n\\begin{proof}\nThe top two squares contain only pullback maps, since $\\overleftarrow{\\delta}_T$ is the pullback map associated to the identity map. It follows that the top two squares commute, since the underlying diagrams of spaces commute. Similarly, up to multiplication by the factor $\\eue(Q_0, \\gamma_2,\\gamma_1)$, the third square is composed entirely of pullbacks in a commutative square of spaces -- note that $\\overleftarrow{\\beta^{-1}}_T$ is defined to just be multiplication by $\\eue(Q_0,\\gamma_2,\\gamma_1)$. The bottom quadrilateral commutes by Proposition \\ref{mixingprop}.\n\\end{proof}\n\\begin{corollary}\n\\label{simplerComult}\nThere is an equality of morphisms\n\\[\n\\Delta=\\eue(Q_0,\\gamma_2,\\gamma_1)\\cdot\\overleftarrow{\\alpha}\\beta\\overleftarrow{\\underline{\\zeta}}\\overleftarrow{\\delta}.\n\\]\n\\end{corollary}\n\\begin{remark}\nComing back to the case $W=0$ considered in the introduction, it is easy to check that the above formula for $\\Delta$ recovers the naive guess for the definition of the coproduct of Section \\ref{NoPot} in the symmetric case.\n\\end{remark}\n\n\n\n\n\\subsection{Proof of the main theorem}\nNow we come to our main theorem regarding the operation $\\Delta$.\n\n\\begin{theorem}\n\\label{comultalg}\nLet $\\gamma\\in\\mathbb{N}^{Q_0}$, and let $\\gamma^0_1,\\gamma^0_2,\\gamma_1,\\gamma_2$ satisfy\n\\begin{align}\n\\label{decompCond}\n\\gamma_1^0+\\gamma_2^0=\\gamma\\\\ \\nonumber\n\\gamma_1+\\gamma_2=\\gamma.\n\\end{align}\nThen the following diagram commutes:\n\\begin{equation}\n\\label{commdiag}\n\\xymatrix{\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma^0_1}\\boxtimes_+^{\\tw} \\mathcal{H}^{\\Sp}_{Q,W,\\gamma^0_2}\\ar[dd]^-{\\sum\\limits_{\\substack{\\gamma'_1+\\gamma'_2=\\gamma^0_1\\\\\\gamma'_3+\\gamma'_4=\\gamma^0_2}}\\Delta_{\\gamma'_1,\\gamma'_2}\\boxtimes_+^{\\tw}\\Delta_{\\gamma'_3,\\gamma'_4}}\\ar[rrr]^-{m}\n&&&\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma}\\ar[dd]^{\\Delta_{\\gamma_1,\\gamma_2}}\n\\\\ \\\\\n{\\sum\\limits_{\\substack{\\gamma'_1+\\gamma'_2=\\gamma^0_1\\\\\\gamma'_3+\\gamma'_4=\\gamma^0_2}}\\left[\n\\begin{matrix}\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma'_1}\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_2'} \\boxtimes_+^{\\tw} \\ldots\\\\ \\ldots \\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_3'}\\boxtimes_+^{\\tw}\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_4'} \n\\end{matrix}\\right]_{\\left\\{(1,2),(3,4)\\right\\}}\n}\n\\ar[rrr]^-{\\tilde{m}^{ 2}}\n&&&\n\\mathcal{H}^{\\Sp}_{Q,W,\\gamma_1}\\tilde{\\boxtimes_+^{\\tw}} \\mathcal{H}^{\\Sp}_{Q,W,\\gamma_2}\n}\n\\end{equation}\nand the operation $\\Delta$ defines a $Q$-localised bialgebra structure on $\\mathcal{H}^{\\Sp}_{Q,W}$ in the category $\\mathcal{C}$. \n\\end{theorem}\n\\begin{proof}\nFor $\\gamma_1,\\gamma_2,\\gamma_1^0,\\gamma_2^0$ as in (\\ref{decompCond}), we define $\\DData_{\\gamma_1,\\gamma_2,\\gamma_1^0,\\gamma_2^0}$ to be the set of $(\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4)\\in(\\mathbb{N}^{Q_0})^4$ such that\n\\begin{align*}\n\\gamma'_1+\\gamma'_2=&\\gamma_1^0\\\\\n\\gamma'_1+\\gamma'_3=&\\gamma_1,\n\\end{align*}\nor if the data $(\\gamma_1,\\gamma_2,\\gamma_1^0,\\gamma_2^0)$ is fixed, we abbreviate $\\DData_{\\gamma_1,\\gamma_2,\\gamma_1^0,\\gamma_2^0}$ to $\\DData$.\n\n\n\\begin{lemma}\n\\label{Dround}\nLet $X$ be a complex algebraic variety carrying an action of $G_{\\gamma}$, let $X^{\\Sp}$ be a $G_{\\gamma}$-equivariant subvariety, and define $V=\\mathrm{H}_{c,T_{\\gamma}}(X^{\\Sp},\\phi_f)^*$. For $\\nabla=(\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4)\\in\\DData$ we consider the diagram\n\\[\n\\xymatrix{\nV^{\\SG_{\\gamma_1^0}\\times\\SG_{\\gamma_2^0}}\\ar[r]^-a\\ar[d]^-{b_{\\nabla}}&V^{\\SG_{\\gamma}}\\ar[d]^-c\\\\\nV^{\\SG_{\\gamma'_1}\\times\\SG_{\\gamma'_2}\\times\\SG_{\\gamma'_3}\\times\\SG_{\\gamma'_4}}\\ar[r]^-{d_{\\nabla}}&V^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\n}\n\\]\nof underlying vector spaces (the maps we define below do not respect the cohomological grading). We define $a$ via the commutativity of the following diagram\n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma_1^0,\\gamma_2^0}}(X,\\phi_f)^*\\ar[d]^{\\cong}\\ar[r]&\\mathrm{H}_{c,G_{\\gamma}}(X,\\phi_f)^*\\ar[d]^{\\cong}\\\\\nV^{\\SG_{\\gamma}}\\ar[r]^-a&V^{\\SG_{\\gamma_1^0}\\times\\SG_{\\gamma_2^0}}\n}\n\\]\nwhere the top map is defined in the same way as $\\underline{\\zeta}$, and the vertical isomorphisms as in Proposition \\ref{invprop}. Similarly we define $c$ in the same way as $\\overleftarrow{\\underline{\\zeta}}$. Likewise we define $d_{\\nabla}$ as the map making the following diagram commute\n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma'_1,\\gamma'_3}\\times G_{\\gamma'_2,\\gamma'_4}}(X^{\\Sp},\\phi_f)\\ar[d]^{\\cong}\\ar[r]& \\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp},\\phi_f)\\ar[d]^{\\cong}\\\\\nV^{\\SG_{\\gamma'_1}\\times\\SG_{\\gamma'_2}\\times\\SG_{\\gamma'_3}\\times\\SG_{\\gamma'_4}}\\ar[r]^-{d_{\\nabla}}&V^{\\SG_{\\gamma_1}\\times\\SG_{\\gamma_2}}\n}\n\\]\nand $b_{\\nabla}$ similarly. Inside $\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(\\mathop{\\rm pt})\\cong\\mathbb{Q}[x^{(1)}_{1,1},\\ldots,x^{(1)}_{n,\\gamma_2(n)},x^{(2)}_{1,1},\\ldots,x^{(2)}_{n,\\gamma_2(n)}]^{\\SG_{\\gamma_2}}$, let $\\mathcal{I}_{\\gamma_1,\\gamma_2}$ be the ideal of functions vanishing when all the $x^{(1)}_{i,\\bullet}$ are evaluated at zero, and all the $x^{(2)}_{i,\\bullet}=\\lambda$, for all $\\lambda\\in\\mathbb{C}$. Then after localising the target at $\\mathcal{I}_{\\gamma_1,\\gamma_2}$, we have the equality of maps\n\\[\n(\\cdot \\eue(Q_0,\\gamma_2,\\gamma_1))\\circ c a=\\sum_{\\nabla\\in\\DData}(-1)^{\\gamma'_2\\cdot\\gamma'_3}d_{\\nabla}\\circ (\\cdot \\eue(Q_0,\\gamma'_2,\\gamma'_1)\\eue(Q_0,\\gamma'_4,\\gamma'_3))\\circ b_{\\nabla}.\n\\]\n\\end{lemma}\n\\begin{proof}\nConsider the Cartesian diagram\n\\[\n\\xymatrix{\n(X^{\\Sp},G_{\\gamma^0_1,\\gamma^0_2})_N\\ar[d]^-{s_N}&Z_N\\ar[d]^-{t_N}\\ar[l]^-{r_N}\\\\\n(X^{\\Sp},G_{\\gamma})_N& (X^{\\Sp},G_{\\gamma_1,\\gamma_2})_N\\ar[l]^-{u_N}\n}\n\\]\nwhere $Z_N$ is the fibre product. We can explicitly describe $Z_N$ as the quotient of the space $\\overline{Z}_N$ by the action of $G_{\\gamma_1,\\gamma_2}$, where $\\overline{Z}_N$ is the space of triples $(x,\\mathbf{f},V)$, where $x\\in X^{\\Sp}$, \n\\[\n\\mathbf{f}=\\left(\\mathbf{f}_{1,1},\\ldots,\\mathbf{f}_{1,\\gamma(1)},\\ldots,\\mathbf{f}_{n,1},\\ldots,\\mathbf{f}_{n,\\gamma(n)}\\right)\\in\\Fr(\\sum_{i\\in Q_0}\\gamma(i),N)\n\\]\nand $V$ is an $i$-tuple of vector subspaces $V_i\\subset \\Span(\\mathbf{f}_{i,1},\\ldots,\\mathbf{f}_{i,\\gamma(i)})$ where $\\dim(V_i)=\\gamma^0_1(i)$. Via the inclusion defined by $\\textbf{f}$ there is a $G_{\\gamma}$-equivariant isomophism $\\overline{Z}_N\\cong \\overline{Z}'_N$ where $\\overline{Z}'_N$ is the space of triples $(x,V',\\mathbf{f})$, where $x$ and $\\mathbf{f}$ are as before, but $V'_i\\subset \\mathbb{C}^{\\gamma_i}$ is a $\\gamma_1^0(i)$-dimensional subspace of a fixed $\\gamma$-dimensional vector space, and $G_{\\gamma_1,\\gamma_2}$ acts via translation of that vector space, i.e. \n\\[\nV'\\in\\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma):=\\prod_{i\\in Q_0}\\mathop{\\rm Gr}\\nolimits(\\gamma_1^0(i),\\gamma(i)). \n\\]\nIn other words we have $Z_N\\cong (X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),G_{\\gamma_1,\\gamma_2})_N$. The map $ca$ is defined by $u^*s_*$, and by Proposition \\ref{ppCor} we can write $u^*s_*=t_*r^*$.\n\nConsider the commutative map of spaces\n\\[\n\\xymatrix{\n(X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),G_{\\gamma_1,\\gamma_2})_N\\ar[d]^{t_N}&\\ar[l](X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),G_{\\gamma_1}\\times G_{\\gamma_2})_N\\ar[d]^{t'_N}\\\\\n(X,G_{\\gamma_1,\\gamma_2})_N&\\ar[l](X,G_{\\gamma_1}\\times G_{\\gamma_2})_N.\n}\n\\]\nWe denote by $\\overline{f}$ the functions on each of these spaces induced by the projection to $X$.\n Then we have a commutative diagram in cohomology\n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma_1,\\gamma_2}}(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),\\phi_{\\overline{f}})^*\\ar[d]^{t_*}\\ar[r]&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),\\phi_{\\overline{f}})^*\\ar[d]^{t'_*}\\\\\n\\mathrm{H}_{c, G_{\\gamma_1,\\gamma_2}}(X^{\\Sp},\\phi_f)^*\\ar[r]&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp},\\phi_{\\overline{f}})^*\n}\n\\]\nwhere the horizontal maps are the ismorphisms induced by pullback along affine fibrations. Define $(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma))^F$ to be the fixed point locus of $X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma)$ under the action of \n\\[\nS^1=\\{z\\in\\mathbb{C}\\textrm{ such that }|z|=1\\}\\subset G_{\\gamma_2}, \n\\]\nembedded via $z\\mapsto (z\\cdot\\id_{\\mathbb{C}^{\\gamma_2(i)}})_{i\\in Q_0}$. Then we have a decomposition\n\\[\n(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma))^F=\\coprod_{(\\gamma'_1,\\ldots,\\gamma'_4)\\in\\DData}X^{\\Sp,F}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)\n\\]\nand so, in particular, an inclusion \n\\[\n(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma))^F\\subset X^{\\Sp}\\times \\coprod_{(\\gamma'_1,\\ldots,\\gamma'_4)\\in\\DData} \\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)\n\\]\nand so the bottom map in the commutative diagram \n\\[\n\\xymatrix{\n\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),\\phi_{\\overline{f}})^*_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\ar[r]^-{\\res'} \\ar[d]^{\\cong}&\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),\\phi_{\\overline{f}}|_{X\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)})^*_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\ar[d]^{\\cong}\n\\\\\n\\mathrm{H}_{G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),E)_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\ar[r] &\\mathrm{H}_{G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),E|_{X\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)})_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\n}\n\\]\nis an isomorphism by \\cite[Thm.6.2]{GKM97}. Here $E:=D(\\phi_{\\overline{f}}\\mathbb{Q}_{X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma)}|_{X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma)})$. Note that since $\\res$ is a pullback, and the space $X^F\\times \\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)$ is disconnected with smooth components of varying dimensions, the map $\\res'$ induces a different degree shift for each component. By definition, the function $\\overline{f}|_{X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma)}$ is given by $f\\boxplus 0$ on $X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma)$. So by the Thom--Sebastiani isomorphism there is a chain of isomorphisms\n\\begin{align*}\n\\phi_{\\overline{f}}|_{X\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)}\\cong&\\phi_f\\boxtimes \\left(\\mathbb{Q}_{\\mathop{\\rm Gr}\\nolimits(\\gamma^0_1,\\gamma)}|_{\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)}\\right)\\\\\n\\cong&\\phi_f\\boxtimes\\mathbb{Q}_{\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)}\\\\\n\\cong&\\phi_{\\overline{f}|_{X\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)}}\n\\end{align*}\nand so we deduce that the map\n\\[\n\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times \\mathop{\\rm Gr}\\nolimits(\\gamma_1^0,\\gamma),\\phi_{\\overline{f}})^*_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\xrightarrow{\\res} \\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2),\\phi_{\\overline{f}})^*_{\\mathcal{I}_{\\gamma_1,\\gamma_2}}\n\\]\nis also an isomorphism. From the commutative diagram\n\\[\n\\xymatrix{\n(X,G_{\\gamma'_1,\\gamma'_3}\\times G_{\\gamma'_2,\\gamma'_4})_N\\ar[rdd]_{p_{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4,N}}\\ar[r]^-{\\cong} &(X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2),G_{\\gamma_1}\\times G_{\\gamma_2})_N\\ar[d]\\\\\n& (X\\times \\mathop{\\rm Gr}\\nolimits(\\gamma^0_1,\\gamma),G_{\\gamma_1}\\times G_{\\gamma_2})_N\\ar[d]\\\\\n&(X,G_{\\gamma_1}\\times G_{\\gamma_2})_N\n}\n\\]\nso we deduce that \n\\begin{align*}\n&t'_{*,\\mathcal{I}_{\\gamma_1,\\gamma_2}}|_{\\res^{-1}(\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)))}\\cdot \\eue(Q_0,\\gamma'_4,\\gamma'_1)\\eue(Q_0,\\gamma'_2,\\gamma'_3)=\\\\&p_{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4,*,\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\res_{\\res^{-1}(\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X^{\\Sp}\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)))}\n\\end{align*}\nsince $\\eue(Q_0,\\gamma'_4,\\gamma'_1)\\eue(Q_0,\\gamma'_2,\\gamma'_3)$ is the Euler characteristic of the normal bundle to $\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)$ inside $\\mathop{\\rm Gr}\\nolimits(\\gamma^0_1,\\gamma)$. It follows that \n\\begin{align*}\nt'_{*,\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\circ(\\cdot \\eue(Q_0,\\gamma_2,\\gamma_1))=&\\sum_{(\\gamma'_1,\\ldots,\\gamma'_4)\\in\\DData} t'_{*,\\mathcal{I}_{\\gamma_1,\\gamma_2}}|_{\\res^{-1}(\\mathrm{H}_{c,G_{\\gamma_1}\\times G_{\\gamma_2}}(X\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_1,\\gamma_1)\\times\\mathop{\\rm Gr}\\nolimits(\\gamma'_3,\\gamma_2)))}\\circ \\\\&(\\cdot\\eue(Q_0,\\gamma'_3,\\gamma'_1)\\eue(Q_0,\\gamma'_4,\\gamma'_1)\\eue(Q_0,\\gamma'_3,\\gamma'_2)\\eue(Q_0,\\gamma'_4,\\gamma'_2))\n\\\\=&(-1)^{\\gamma'_2\\cdot\\gamma'_3}\\sum_{(\\gamma'_1,\\ldots,\\gamma'_4)\\in\\DData}p_{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4,*,\\mathcal{I}_{\\gamma_1,\\gamma_2}}\\circ (\\cdot \\eue(Q_0,\\gamma'_1,\\gamma'_2)\\eue(Q_0,\\gamma'_3,\\gamma'_4)).\n\\end{align*}\nHere we use the identity\n\\[\n\\eue(Q_0,\\gamma'_2,\\gamma'_3)=(-1)^{\\gamma'_2\\cdot\\gamma'_3}\\eue(Q_0,\\gamma'_3,\\gamma'_2).\n\\]\nSince $p_{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}$ is the map inducing $b_{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}$, the desired equality follows.\n\\end{proof}\n\nTo finish the proof of Theorem \\ref{comultalg} we introduce some more notation. Given $\\nabla=(\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4)\\in\\DData_{\\gamma_1,\\gamma_2,\\gamma_1^0,\\gamma_2^0}$ and $D$ an array of dots in a four by four grid, we define $M^{\\Sp}_{\\nabla,D}\\subset M^{\\Sp}_{\\gamma}$ to be the subspace of $\\rho\\in M^{\\Sp}_{\\gamma}$ such that for every $a\\in Q_1$, \n\\[\n\\rho\\left(\\mathbb{C}^{\\gamma'_c(s(a))}\\right)\\subset\\bigoplus_{\\substack{c'\\textrm{ such that }D\\textrm{ has}\\\\ \\textrm{a dot in position }(c',c)}}\\mathbb{C}^{\\gamma'_{c'}(t(a))}.\n\\]\nSo for example\n\\[\nM^{\\Sp}_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}\n\\]\nis the subspace of $M^{\\Sp}_{\\gamma}$ of representations $\\rho$ such that \n\\begin{align*}\n\\rho (\\mathbb{C}^{\\gamma'_1})\\subset &\\mathbb{C}^{\\gamma'_1}\\oplus \\mathbb{C}^{\\gamma'_2}\\\\\n\\rho(\\mathbb{C}^{\\gamma'_2})\\subset & \\mathbb{C}^{\\gamma'_2}\\\\\n\\rho(\\mathbb{C}^{\\gamma'_4})\\subset &\\mathbb{C}^{\\gamma'_2}\\oplus \\mathbb{C}^{\\gamma'_4}.\n\\end{align*}\nWe denote by $W_{\\nabla,D}$ the restriction of $\\tr(W)$ to $M_{\\nabla,D}$, and define\n\\[\nV_{\\nabla,D}=\\mathrm{H}_{c,T_{\\gamma}}(M^{\\Sp}_{\\nabla,D},W_{\\nabla,D})^*.\n\\]\nDepending on the array $D$, the space $V_{\\nabla,D}$ carries an action of a group $G\\subset \\SG_{\\gamma}$ where $G$ contains $\\SG_{\\gamma'_1}\\times\\SG_{\\gamma'_2}\\times\\SG_{\\gamma'_3}\\times\\SG_{\\gamma'_4}$. When restricting to the invariant part of the $G$-action, for $G=\\SG_{\\gamma''_1}\\times\\ldots\\times\\SG_{\\gamma''_c}$, we abbreviate\n\\[\nV_{\\nabla,D}^{\\SG_{\\gamma''_1}\\times\\ldots\\times\\SG_{\\gamma''_c}}\n\\]\nto \n\\[\nV_{\\nabla,D}^{\\gamma''_1,\\ldots,\\gamma''_c}.\n\\]\nFinally, if the element $\\nabla$ is missing from the notation, but elements $\\gamma'_1,\\ldots,\\gamma'_4$ appear in superscripts, we take a direct sum over all choices of $\\nabla\\in\\Dec_{\\gamma_1,\\gamma_2,\\gamma^0_1\\gamma^0_2}$.\n\n\\begin{lemma}\n\\label{LemMod}\nConsider the commutative diagram\n\\[\n\\xymatrix{\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar[r]^a\\ar[d]^b&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[d]^c\n\\\\\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar[r]^d&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n}\n\\]\nwhere $a$ and $d$ are given by pushforward along inclusions, and $b$ and $c$ are given by pullbacks. Then $c a=d b(\\cdot \\eue(Q_1,\\gamma'_2,\\gamma'_3))$.\n\\end{lemma}\n\\begin{proof}\nWe can extend the diagram as follows.\n\\[\n\\xymatrix{\n&&V_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar@\/^2pc\/[ddl]^c\\ar@\/^1pc\/[dl]_{i^*}\n\\\\\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar@\/^2pc\/[urr]^a\\ar[d]^{b}\\ar[r]^{a'}&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[d]^{c'}\\ar@\/^1pc\/[ur]^{i_*}\n\\\\\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar[r]^d&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n}\n\\]\nwhere all maps are still either pullback maps or pushforward maps along the obvious inclusions, and the inner square is the Cartesian product of a transversal intersection of manifolds. Then by Corollary \\ref{ppCor},\n\\[\nc' a'=d b,\n\\]\nand by Proposition \\ref{maneq}, \n\\[\ni^*i_*=\\cdot\\eue(Q_1,\\gamma'_2,\\gamma'_3).\n\\]\nSince $a,i_*,a'$ are all pushforward maps, it follows that $a=a'i_*$. Similarly we deduce that $c=c'i^*$, and the lemma follows.\n\n\n\\end{proof}\n\\begin{lemma}\nConsider the diagram\n\\label{LemMod2}\n\\[\n\\xymatrix{\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar[r]^-a\\ar[d]^b&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[d]^c\n\\\\\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture},(1,2),(3,4)}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n\\ar[r]^-d&\nV_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture},(1,2),(3,4),(2,3)}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\n}\n\\]\nwhere all the maps are induced by pushforward or pullback along affine fibrations, so we have localised the targets of the vertical maps. Then $ca=db\\circ(\\cdot \\eue(Q_1,\\gamma'_2,\\gamma'_3)$\n\\end{lemma}\n\\begin{proof}\nThe proof of the lemma is the same as the proof of Lemma \\ref{LemMod}, using the other half of Proposition \\ref{maneq}, and using that in the following diagram of affine fibrations\n\\[\n\\xymatrix{\n&&\nM_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}\\ar[dl]\\ar[dll]\\ar[ddl]\n\\\\\nM_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}\\ar[d]\n&\nM_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\end{picture}}\\ar[l]\\ar[d]\n\\\\\nM_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture}}\n&\nM_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture}}\\ar[l]\n}\n\\]\nthe inner square is Cartesian, and the maps $a,b,c,d$ are given by pullbacks or pushforwards induced by the morphisms in the outer square.\n\\end{proof}\nWe can summarise Lemmas \\ref{Dround}, \\ref{LemMod} and \\ref{LemMod2} in the following diagram\n\\begin{equation}\n\\label{bigDiag}\n\\xymatrix{\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\end{picture}}^{\\gamma^0_1,\\gamma^0_2}\\ar[r]\\ar[d]\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\end{picture}}^{\\gamma^0_1,\\gamma_2^0}\\ar[r]\\ar[d]\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma^0_1,\\gamma^0_2}\\ar[r]\\ar[d]\\save[]+<1.5cm,-.9cm>*\\txt{Lemma \\ref{Dround}}\\restore\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma}\\ar[d]\n\\\\\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[r]\\ar[d]\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[r]\\ar[d]\\save[]+<1.7cm,-.9cm>*\\txt{Lemma \\ref{LemMod}}\\restore\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\save[]+<1.5cm,-.9cm>*\\txt{Lemma \\ref{LemCatch}}\\restore\\ar[r]\\ar[d]\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma_1,\\gamma_2}\\ar[d]\n\\\\\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[r]\\ar[d]\\save[]+<1.7cm,-.9cm>*\\txt{Lemma \\ref{LemMod2}}\\restore\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[r]\\ar[d]\n&\nV^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\\ar[r]\\ar[d]\n&\nV^{\\gamma_1,\\gamma_2}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\\ar[d]\n\\\\\nV^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture},(1,2),(3,4)}\\ar[r]\n&\nV^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,1){\\circle*{2}}\\end{picture},(1,2),(3,4),(2,3)}\\ar[r]\n&\nV^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture},\\substack{(1,2),(3,4),\\\\(2,3),(1,4)}}\\ar[r]\n&\nV^{\\gamma_1,\\gamma_2}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture},(1,2)}.\n}\n\\end{equation}\n\\begin{lemma}\n\\label{LemCatch}\nThe square formed by the perimeter of the diagram \n\\[\n\\xymatrix{\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma^0_1,\\gamma^0_2}\\ar[r]\\ar[d]\\\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma}\\ar[d]\n\\\\\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}\\ar[r]\\ar[d]\n&\nV_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(4,10){\\circle*{2}}\\put(4,4){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(10,4){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(10,10){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}^{\\gamma_1,\\gamma_2}\\ar[d]\n\\\\\nV^{\\gamma'_1,\\gamma'_2,\\gamma'_3,\\gamma'_4}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\\ar[r]\n&\nV^{\\gamma_1,\\gamma_2}_{\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\n}\n\\]\ncommutes.\n\\end{lemma}\n\\begin{proof}\nThe bottom of the two squares commutes by Corollary \\ref{ppCor}, since the horizontal maps are given by pushforward along smooth submersions. By Lemma \\ref{Dround}, the top square commutes after localising with respect to $\\mathcal{I}_{\\gamma_1,\\gamma_2}$, and so the lemma will follow from the claim that the localisation map\n\\[\nV^{\\gamma_1,\\gamma_2}_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\\rightarrow V^{\\gamma_1,\\gamma_2}_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture},\\mathcal{I}_{\\gamma_1,\\gamma_2}}\n\\]\nis an injection. For this, it suffices to show that if \n\\[\n\\alpha\\in V^{\\gamma_1,\\gamma_2}_{\\nabla,\\begin{picture}(12,14)\\put(1,10){\\circle*{2}}\\put(4,7){\\circle*{2}}\\put(1,7){\\circle*{2}}\\put(1,1){\\circle*{2}}\\put(7,4){\\circle*{2}}\\put(7,10){\\circle*{2}}\\put(10,1){\\circle*{2}}\\put(7,1){\\circle*{2}}\\put(10,7){\\circle*{2}}\\put(7,7){\\circle*{2}}\\put(1,4){\\circle*{2}}\\put(4,1){\\circle*{2}}\\end{picture}}\n\\]\nand \n\\[\nz\\in \\mathrm{H}_{G_{\\gamma_1^0}\\times G_{\\gamma_2^0}}(\\mathop{\\rm pt})\\setminus \\mathcal{I}_{\\gamma_1,\\gamma_2}, \n\\]\nthen $z\\cdot \\alpha\\neq 0$.\nFor this the argument is as in the proof of Proposition \\ref{ABprop}.\n\\end{proof}\n\\begin{lemma}\nAll of the small squares in diagram (\\ref{bigDiag}) that are not marked with lemmas commute.\n\\end{lemma}\n\\begin{proof}\nEach of the squares is either entirely composed of pullback maps or pushforward maps along a commutative diagram, in which case commutativity is trivial, or else commutativity is an application of Proposition \\ref{mixingprop}.\n\\end{proof}\n\nWe now have all of the components in place to prove Theorem \\ref{comultalg}. If we let $e_{\\nabla}$ be the composition of the leftmost vertical maps, for a fixed summand corresponding to $\\nabla\\in\\DData$, then by the definition of $\\Delta$ and Corollary \\ref{simplerComult}, we have the identity\n\\[\n\\Delta_{\\gamma'_1,\\gamma'_2}\\otimes\\Delta_{\\gamma'_3,\\gamma'_4}=(\\cdot \\eue(Q_0,\\gamma'_2,\\gamma'_1)\\eue(Q_0,\\gamma'_4,\\gamma'_3))\\circ e_{\\nabla}\n\\]\nand similarly, if we let $f$ be the composition of the rightmost vertical maps in (\\ref{bigDiag}) we have\n\\[\n\\Delta_{\\gamma_1,\\gamma_2}=(\\cdot\\eue(Q_0,\\gamma_2,\\gamma_1))\\circ f.\n\\]\nOn the other hand, by Lemma \\ref{LemCatch}, up to the sign $(-1)^{\\gamma'_2\\cdot\\gamma'_3}$ these are exactly the factors introduced by the noncommutativity of the top two squares in the rightmost column of (\\ref{bigDiag}). The remaining failure of the diagram to commute is described by Lemmas \\ref{LemMod} and \\ref{LemMod2}, so that the overall noncommutativity of the diagram is given by the factor $\\tilde{\\eue}_{\\gamma'_2,\\gamma'_3}$, as defined in (\\ref{corr_term}) in the definition of the symmetrising morphism $\\tilde{\\sw}_{\\gamma'_2,\\gamma'_3}$ and so the compatibility condition of diagram (\\ref{commdiagB}) is satisfied.\n\nThe injectivity of localisation condition, and the crosslinearity condition, contained in the definition of a $Q$-localised bialgebra, are the content of Propositions \\ref{ABprop} and \\ref{coCross} respectively, and we are done.\n\\end{proof}\n\\section{Examples}\n\\subsection{The Jordan quiver with potential}\nLet $Q_{\\Jor}$ be the quiver with one vertex and one loop, which we label $X$. Let $d\\in\\mathbb{Z}_{\\geq 1}$, we set $W_d=X^{d+1}$. The motivic Donaldson--Thomas theory of the pair $(Q_{\\Jor},W_d)$ was studied in the paper \\cite{DM11}, and the motivic Donaldson--Thomas invariants were calculated there: we have\n\\[\n\\Omega_{Q_{\\Jor},W_d,n}=\\begin{cases} \\mathrm{H}_{c}(\\mathbb{A}^1,\\phi_{x^{d+1}})&\\textrm{if }n=1\\\\0&\\textrm{otherwise.}\\end{cases}\n\\]\nBy our main theorem, the algebra $\\mathcal{H}_{Q_{\\Jor},W_d}$ carries a localised coproduct\n\\[\n\\Delta:\\mathcal{H}_{Q_{\\Jor},W_d}\\tilde{\\boxtimes}^{\\sw}_{+}\\mathcal{H}_{Q_{\\Jor},W_d}.\n\\]\nHowever in this special case things are a little simpler than in general. In fact the map $\\Delta$ lifts to a map\n\\[\n\\Delta:\\mathcal{H}_{Q_{\\Jor},W_d}\\otimes\\mathcal{H}_{Q_{\\Jor},W_d}.\n\\]\nas the division factor in the definition of $\\overleftarrow{\\alpha}$ is exactly the factor by which we multiply $\\beta$ in the definition of $\\Delta$. Secondly, the multiplication factor in the symmetrising morphism $\\tilde{\\sw}$ is equal to one, i.e. we consider the target of $\\Delta$ in the usual symmetric monoidal category $\\mathop{\\mathbf{MMHS}}\\nolimits_{\\mathbb{Z}}$. In other words, $\\mathcal{H}_{Q_{\\Jor},W_d}$ is a Hopf algebra (the construction of a unique antipode is formal, using that $\\mathcal{H}_{Q_{\\Jor},W_d}$ is a connected algebra). Let $\\mathfrak{g}\\subset\\mathcal{H}_{Q_{\\Jor},W_d}$ be the subspace of primitive elements, then by general theory $\\mathfrak{g}$ is a Lie algebra, with Lie bracket given by the commutator, and the natural map\n\\[\n\\iota:\\mathcal{U}(\\mathfrak{g})\\rightarrow\\mathcal{H}_{Q_{\\Jor},W_d}\n\\] \nis an injection.\n\\begin{proposition}\nThe following are equivalent:\n\\begin{enumerate}\n\\item\nThe element $\\mathcal{H}_{Q_{\\Jor},W_d}\\in\\Ob(D(\\mathop{\\mathbf{MMHS}}\\nolimits))$ is pure, in the sense that the $i$th cohomologically graded piece is pure of weight $i$,\n\\item\nThe Lie algebra $\\mathfrak{g}$ lives entirely in degree $1$, with respect to the $\\mathbb{Z}^{Q_0}$-grading, and $\\iota$ is an isomorphism ${\\rm Sym}(\\mathcal{H}_{Q_{\\Jor},W,1})\\rightarrow\\mathcal{H}_{Q_{\\Jor},W_d}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\\begin{itemize}\n\\item[$1\\rightarrow 2$:] If $\\mathcal{H}_{Q_{\\Jor},W_d}$ is pure, it follows that $\\mathfrak{g}$ is too, and hence so is $\\mathcal{U}(\\mathfrak{g})$. Clearly the whole of $\\mathcal{H}_{Q_{\\Jor},W_d,1}$ is primitive. By the main result of \\cite{DM11}, the algebra $A\\subset \\mathcal{U}(\\mathfrak{g})$ generated by $\\mathcal{H}_{Q_{\\Jor},W_d,1}$ has the same class in $\\KK(\\mathop{\\rm MMHM}\\nolimits_{\\mathbb{Z}^{Q_0}})$ as the target of $\\iota$, and so it follows from purity that $\\iota$ is an isomorphism when restricted to $A$, and so $A=\\mathcal{U}(\\mathfrak{g})$ since $\\iota$ is injective. Since $\\mathfrak{g}$ is concentrated entirely in $\\mathbb{Z}^{Q_0}$-degree one, it is Abelian, and so $\\iota$ becomes the isomorphism ${\\rm Sym}(\\mathcal{H}_{Q_{\\Jor},W_d,1})\\rightarrow\\mathcal{H}_{Q_{\\Jor},W_d}$.\n\\item[$2\\rightarrow 1$:] The monodromic mixed Hodge structure on \n\\begin{align*}\n\\mathcal{H}_{Q_{\\Jor},W_d,1}=&\\mathrm{H}_{c,\\mathbb{C}^*}(\\mathbb{A}^1,\\phi_{x^{d+1}})^*\\\\\n\\cong&\\mathrm{H}_{c}(\\mathbb{A}^1,\\phi_{x^{d+1}})^*\\otimes\\mathrm{H}_{\\mathbb{C}^*}(\\mathop{\\rm pt})\n\\end{align*} is pure, and so the symmetric algebra generated by it is pure too.\n\\end{itemize}\n\\end{proof}\nPurity of $\\mathcal{H}_{Q_{\\Jor},W_d}$ is quite easy to prove, using the representation theory of $\\mathcal{H}_{Q,W}$, for general triples $(Q,W,\\Sp)$ -- this representation theory is worked out in some detail in \\cite{DaMe15b}. But for our special choice of quiver with potential, the situation is easier to describe. For $n,f\\in\\mathbb{N}$, define $V_{n,f}=\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^n,\\mathbb{C}^n)\\times\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^n,\\mathbb{C}^f)$. We think of $V_{n,f}$ as a space of representations of the coframed Jordan quiver, of dimension vector $(n,f)$. The space $V_{n,f}$ carries the action of $G_{n}$ via change of basis, as always. Let $V_{n,f}^{st}\\subset V_{n,f}$ be the subset of representations $\\rho$ such that the there is no nonzero subrepresentation $\\rho'\\subset\\rho$ supported on the original quiver $Q_{\\Jor}$. We have a chain of $G_n$-equivariant inclusions\n\\[\nM_n\\times\\Fr(n,f))\\subset V^{st}_{n,f}\\subset V_{n,f}.\n\\]\nAs we let $f$ become very large, the codimension of $\\left(V^{st}_{n,f}\/G_n\\setminus (M_n\\times\\Fr(n,f))\/G_n\\right)$ becomes very large too, and it follows from the fact that $\\phi_{\\tr(W)}[fn]$ is a perverse sheaf on $V^{st}_{n,f}\/G_n$ that the map in compactly supported cohomology \n\\[\n\\mathrm{H}^{i+fn}_{c}( (M_n,G_n)_f,\\phi_{\\tr(W_d)_f})\\rightarrow \\mathrm{H}^{i+fn}_c(V^{st}_{n,f}\/G_n,\\phi_{\\tr(W_d)_f})\n\\]\nis an isomorphism for fixed $i$ and large $f$, where $\\tr(W_d)_f$ is the function on $V_{n,f}^{st}\/G_n$ induced by $\\tr(W_d)$. Now the map $p_f:V^{st}_{n,f}\/G_n\\rightarrow \\mathbb{A}^n$ to the affine quotient is proper, and so there is a natural isomorphism\n\\[\np_{f,*}\\phi_{\\tr(W_d)_f}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}\\cong\\phi_{g}p_{f,*}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}\n\\]\nwhere $g:\\mathbb{A}^n\\rightarrow\\mathbb{A}^1$ is the map induced by $\\tr(W_d)$. In addition, the complex of mixed Hodge modules $p_{f,*}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}$ is pure by Saito's version of the decomposition theorem \\cite{Sa88}, in the sense that $\\mathcal{H}^i(p_{f,*}\\mathbb{Q}_{V^{st}_{n,f}\/G_n})$ is pure of weight $i$. Furthermore the support of $\\phi_gp_{f,*}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}$ is the origin $0\\in\\mathbb{A}^1$; this follows from the description of the support of $\\phi_{\\tr(W_d)_f}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}$ in terms of the Jacobi algebra for the pair $(Q_{\\Jor},W_d)$ (see Remark \\ref{JacRem}) since all representations $\\rho$ of the Jacobi algebra send $X$ to a nilpotent endomorphism, and by \\cite[Thm.1]{LBP90} the functions on the affinization $\\mathbb{A}^n$ are given by evaluating $\\tr(\\rho(X)^i)$ for $i\\in\\mathbb{N}$. In particular the support is proper, and it follows as in \\cite[Cor.3.2]{DMSS12}, after passing to the intermediate extension of $p_{f,*}\\mathbb{Q}_{V^{st}_{n,f}\/G_n}$ to a $\\mathbb{C}^*$-equivariant relative compactification of the morphism $\\mathbb{A}^n\\xrightarrow{g}\\mathbb{A}^1$, that $\\mathrm{H}_c(V^{st}_{n,f}\/G_n,\\phi_{\\tr(W)_f})\\state{{fn}}$ is pure as a monodromic mixed Hodge structure. We have proved the following result, which one may already find without proof in \\cite[Sec.2.4]{So14}.\n\\begin{theorem}\nThe algebra $\\mathcal{H}_{Q_{\\Jor},W}$ is supercommutative, and there is an isomorphism of algebras\n\\begin{align*}\n\\mathcal{H}_{Q_{\\Jor},W}\\cong&\\bigwedge \\mathrm{H}(\\mathbb{A}^1,\\phi_{x^{d+1}})\\otimes\\mathrm{H}_{\\mathbb{C}^*}(\\mathop{\\rm pt})[-1]\n\\\\\n\\cong& \\bigwedge_{r=1}^d\\bigwedge[u_{r,1},u_{r,2},\\ldots],\n\\end{align*}\nwhere $u_{r,n}$ lives in cohomological degree $2n-1$.\n\\end{theorem}\n\n\n\\subsection{Twisted and untwisted character varieties}\n\\label{charSec}\nLet $\\Sigma_g$ denote the genus $g$ topological Riemann surface. We define\n\\[\n\\Rep_m(\\Sigma_g):=\\{A_1,\\ldots,A_g,B_1,\\ldots,B_g\\in\\Gl_{\\mathbb{C}}(m)|\\prod (A_i,B_i)=\\id_{m\\times m}\\},\n\\]\nwhere $(A_i,B_i)$ is the group theoretic commutator. This variety is a space of homomorphisms $\\pi_1(\\Sigma_g)\\rightarrow \\Gl_{\\mathbb{C}}(m)$, or representations of $\\pi_1(\\Sigma_g)$, and the stack theoretic quotient $[\\Rep_m(\\Sigma_g)\/\\Gl_{\\mathbb{C}}(m)]$ is the stack of $m$ dimensional $\\pi_1(\\Sigma_g)$ representations. Let $\\zeta_m$ be a primitive $m$th root of unity. We will also consider the twisted counterpart of this variety\n\\[\n\\Rep^{\\zeta_m}_m(\\Sigma_g):=\\{A_1,\\ldots,A_g,B_1,\\ldots,B_g|\\prod(A_i,B_i)=\\zeta_m\\id_{m\\times m}\\}.\n\\]\nBy \\cite[Cor.2.2.4]{HLRV13}, up to isomorphisms in cohomology lifting to isomorphisms of Hodge structure, it does not matter which primitive $m$th root of unity we pick. We will build a CoHA $\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}}$ such that there are natural isomorphisms\n\\[\n\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g},(m,m,m,m)}\\cong \\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_g),\\mathbb{Q})\\state{(1-g)m^2}^*,\n\\]\nand conjectural isomorphisms\n\\[\n\\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_g),\\mathbb{Q})\\state{(1-g)m^2}^*\\cong{\\rm Sym}\\left(\\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep^{\\zeta_m}_m(\\Sigma_g),\\mathbb{Q})\\state{(1-g)m^2}^*\\right)\n\\]\nat the level of $\\mathbb{N}$-graded mixed Hodge structures, where $\\mathbb{N}$ keeps track of the dimension of the $\\mathbb{C}[\\pi_1(\\Sigma_g)]$-reprsentation. By \\cite[Cor.2.2.7]{HLRV13} $\\Rep^{\\zeta_m}_m(\\Sigma_g)$ is acted on freely by ${\\rm PGL}_{\\mathbb{C}}(m)$, so there is an isomorphism \n\\[\n\\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep^{\\zeta_m}_m(\\Sigma_g),\\mathbb{Q})\\state{(1-g)m^2}^*\\cong\\mathrm{H}_c(\\Rep^{\\zeta_m}_m(\\Sigma_g)\/{\\rm PGL}_{\\mathbb{C}}(m),\\mathbb{Q})\\state{(1-g)m^2}^*[u]\n\\]\nwhere $u$ is the degree 2 generator of $\\mathrm{H}_{\\mathbb{C}^*}(\\mathop{\\rm pt},\\mathbb{Q})$. In the language of quantum enveloping algebras, we conjecture that there are isomorphisms\n\\[\n\\mathfrak{g}_{\\prim,(m,m,m,m)}\\cong\\mathrm{H}_c(\\Rep^{\\zeta_m}_m(\\Sigma_g)\/{\\rm PGL}_{\\mathbb{C}}(m),\\mathbb{Q})\\state{(1-g)m^2}^*\n\\]\nbetween the space of primitive generators of a PBW basis for a cocommutative deformation of $\\mathcal{H}_{Q_{\\Sigma_2},W_{\\Sigma_2}}$ and the dual compactly supported cohomology of the twisted character varieties for $\\Sigma_g$, where the primitive generators are defined to be those that generate the other generators under the action of multiplication by $u$. Since by \\cite[Thm.2.2.5]{HLRV13} the twisted character varieties are smooth, we may alternatively restate this as an isomorphism between the primitive generators of $\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}}$ and the shifted cohomology of the twisted character varieties.\n\n\\begin{example}\nFor ease of exposition we consider only the Riemann surface of genus 2, but everything generalises in a way that is hopefully obvious. We break the surface $\\Sigma_2$ into 4 tiles. The front two tiles are as drawn in black in Figure \\ref{g2front}, they are formed from the upper half of the figure glued to the bottom half along the dashed line. \n\\begin{figure}\n\\caption{Tiling of $\\Sigma_2$ seen from the front}\n\\includegraphics{genus2top.eps}\n\\label{g2front}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Tiling of $\\Sigma_2$ seen from the back}\n\\includegraphics{genus2back.eps}\n\\label{g2back}\n\\end{figure}\nThe back two tiles are as drawn in Figure \\ref{g2back}, they are again formed from the top half of the figure glued to the bottom half along the dashed line. The back two tiles are glued to the front two by identifying the solid black lines of Figure \\ref{g2front} with the solid black lines of Figure \\ref{g2back}. We have drawn, in red and blue, the dual quiver to this tiling, this will be our quiver $Q_{\\Sigma_2}$, which we reproduce below:\n\\[\n\\xymatrix{\\\\\n\\bullet\\ar@\/_1.5pc\/@[blue][rrr]_a\\ar@[blue][rrr]_b\\ar@\/^1.5pc\/@[blue][rrr]_c &&&\\bullet \\ar@\/_1.5pc\/@[red][ddd]^d\\ar@\/^1.5pc\/@[red][ddd]^f \\ar@[red][ddd]^e\n\\\\ \\\\\n\\\\\\bullet \\ar@\/_1.5pc\/@[red][uuu]^l\\ar@\/^1.5pc\/@[red][uuu]^j \\ar@[red][uuu]^k&&&\\bullet \\ar@\/_1.5pc\/@[red][lll]^g\\ar@[red][lll]^h\\ar@\/^1.5pc\/@[red][lll]^i\n}\n\\]\nWe consider the following element of $\\mathbb{C}Q_{\\Sigma_2}\/[\\mathbb{C}Q_{\\Sigma_2},\\mathbb{C}Q_{\\Sigma_2}]$:\n\\[\nW_{\\Sigma_2}:=lgfa-jgda+jhdb-kheb+kiec-lifc.\n\\]\nThe recipe for this quiver with potential is as follows - in the literature it is called the QP associated to a brane tiling of a surface, see e.g. \\cite{longout} for a detailed reference, \\cite{Rhombi} for the Physics background, or \\cite{MR} or \\cite{Dav08} for the Mathematics background. To a tiling $\\Delta$ of a Riemann surface, the 1-skeleton of which is given the structure of a bipartite graph, the associated quiver is just the dual quiver, as above, oriented so that the arrows go clockwise around the black vertices. The potential is given by taking the alternating sum\n\\[\nW_{\\Delta}:=\\sum_{v\\in\\Delta_0|v\\text{ is white}}l_v-\\sum_{v\\in\\Delta_0|v\\text{ is black}}l_v,\n\\]\nwhere $l_v$ is the shortest cycle going around the vertex $v$ in the dual quiver.\n\nReturning to our special case, we define $M^{\\Sp}_{\\gamma}\\subset M_{\\gamma}$ by the condition that every red arrow is sent to an isomorphism, and since an upper block triangular matrix is invertible if and only if its diagonal blocks are, we deduce that these $M^{\\Sp}_{Q_{\\Sigma_2},\\gamma}$ satisfy Assumption \\ref{closed_under}.\n\nThe QP $(W_{\\Sigma_2},W_{\\Sigma_2})$ admits a cut in the sense of Section \\ref{2dCOHA}, given by setting $S=\\{a,b,c\\}$, and the moduli spaces $M^{\\Sp}_{Q_{\\Sigma_2},\\gamma}$ satisfy Assumption \\ref{ass1}, so that we have an isomorphism in cohomology\n\\[\n\\mathrm{H}_{c,G_{\\gamma}}(M^{\\Sp}_{Q_{\\Sigma_2},\\gamma},\\phi_{\\tr(W_{\\Sigma_2})_{\\gamma}})\\cong\\mathrm{H}_{c,G_{\\gamma}}(\\overline{Z}_{\\gamma},\\mathbb{Q})\n\\]\nwhere $\\overline{Z}_{\\gamma}$ is the space of representations of $Q_{\\Sigma_2}$ such that all red arrows are sent to isomorphisms, and the relations\n\\begin{align}\n\\label{g2rels}\n\\partial W_{\\Sigma_2}\/\\partial a=lgf-jgd=0\\\\\n\\nonumber\n\\partial W_{\\Sigma_2}\/\\partial b=jhd-khe=0\\\\\n\\nonumber\n\\partial W_{\\Sigma_2}\/\\partial c=kie-lif=0\n\\end{align}\nare satisfied. Let $Z_{\\gamma}$ be the space of representations of the quiver $Q'$, obtained by deleting arrows $a$, $b$ and $c$, still satisfying the relations (\\ref{g2rels}). Up to gauge transformation we may assume that $d$, $g$ and $j$ are all the identity matrix, and consider $[Z_{\\gamma}\/G_{\\gamma}]$ as the stack of representations of the 6 loop quiver algebra with loops labelled by $e$, $f$, $h$, $i$, $k$ and $l$, satisfying the relations\n\\begin{align*}\nfl=1\\\\\nh=khe\\\\\nlif=kie\n\\end{align*}\nsuch that all arrows are sent to invertible matrices. Substituting $l=f^{-1}$ and $k=he^{-1}h^{-1}$, we deduce that $Z_{\\gamma}$ is isomorphic to the stack of $m$-dimensional representations of the 4 loop quiver, with loops labelled $f,h,e,i$, satisfying the one relation\n\\[\nhe^{-1}h^{-1}ie=f^{-1}if,\n\\]\nwhich becomes\n\\[\nhe^{-1}h^{-1}e=i^{-1}f^{-1}if\n\\]\nafter the substitution $h\\mapsto ih$. In other words, $[Z_{\\gamma}\/G_{\\gamma}]$ is isomorphic to the stack $[\\Rep_m(\\Sigma_2)\/\\Gl_{\\mathbb{C}}(m)]$ of $m$-dimensional representations of $\\pi_1(\\Sigma_2)$.\n\nVia the affine fibration $\\overline{Z}_{\\gamma}\\rightarrow Z_{\\gamma}$ we obtain, by Theorem \\ref{eqred}, an isomorphism of mixed Hodge structures\n\\[\n\\mathrm{H}_{c,G_{\\gamma}}(M_{Q_{\\Sigma_2},W_{\\Sigma_2}}^{\\Sp},\\phi_{\\tr(W_{\\Sigma_2})_{\\gamma}})\\cong \\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_2),\\mathbb{Q})\\state{{-3m^2}}.\n\\]\nSince $\\chi((m,m,m,m),(m,m,m,m))=-8m^2$ we deduce that\n\\[\n\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_2},W_{\\Sigma_2},(m,m,m,m)}\\cong \\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_2),\\mathbb{Q})^*\\state{{-m^2}}.\n\\]\n\\end{example}\n\\bigbreak\nGeneralising the above construction we have \n\\[\n\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g},(m,m,m,m)}\\cong \\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_g),\\mathbb{Q})^*\\state{{(1-g)m^2}}.\n\\]\nFrom Theorem \\ref{comultalg} we deduce the following theorem.\n\\begin{theorem}\nThe graded mixed Hodge structure\n\\[\n\\bigoplus_{n\\in\\mathbb{N}}\\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m(\\Sigma_g),\\mathbb{Q})^*\\state{{(1-g)m^2}}\n\\]\ncarries the structure of a $Q$-localised bialgebra in the category of mixed Hodge structures.\n\\end{theorem}\n\nWe finish by returning to the conjectural form for the generators of $\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}}$. \n\\begin{conjecture}\nThere is a filtration $F$ on $\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}}$ such that $\\mathop{\\rm Gr}\\nolimits_F(\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}})\\cong \\mathcal{U}(\\mathfrak{g}[u])$ for $\\mathfrak{g}$ a $\\mathbb{Z}^{Q_{\\Sigma_g,0}}$-graded Lie algebra, and there are isomorphisms in $\\mathop{\\mathbf{MHS}}\\nolimits$\n\\[\n\\mathfrak{g}_{(m,m,m,m)}\\cong\\mathrm{H}_{c}(\\Rep_m^{\\zeta_m}(\\Sigma_g)\/{\\rm PGL},\\mathbb{Q})^*\\state{(1-g)m^2}.\n\\]\n\\end{conjecture}\n\nThe filtration $F$ will be constructed for general triples $(Q,W,\\Sp)$ in the paper \\cite{DaMe15b}. The evidence for this conjecture comes from taking weight polynomials (E polynomials, in the terminology of \\cite{HLRV13}). In \\cite{HLRV13} the following calculation is made:\n\\[\n\\chi_q(\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}})=\\prod(1-x_m)^{\\chi_q\\big(\\mathrm{H}_{c,\\Gl_{\\mathbb{C}}(m)}(\\Rep_m^{\\zeta_m}(\\Sigma_g),\\mathbb{Q})^*\\state{(1-g)m^2}\\big)},\n\\]\nwhich means that if we have $\\mathcal{H}^{\\Sp}_{Q_{\\Sigma_g},W_{\\Sigma_g}}\\cong \\mathcal{U}(\\mathfrak{g}[u])$ then the $(m,m,m,m)$th graded piece of the Lie algebra $\\mathfrak{g}$ has the same weight polynomial as the cohomology of the twisted character variety $\\Rep^{\\zeta_m}_m(\\Sigma_g)$, up to a Tate twist.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.}\n\\label{s_sample}\n In all QSO samples there is concern that selection effects are present and\nsignificant, particularly in whether whole classes of objects are not included, or even\nknown. This study aims at characterizing a population of\nobjects with rising\nfluxes at UV observed wavelengths.\nFollowing our work of classification of UV sources from the GALEX \\footnote{\nThe {\\it Galaxy Evolution Explorer}, GALEX (Martin et al. 2005), is a NASA Small Explorer \nperforming imaging surveys of the sky in two UV bands simultaneously:\nFUV (1344 - 1786 \\AA, $\\lambda$$_{eff}$ = 1528 {\\AA})\nand NUV (1771 - 2831 \\AA, $\\lambda$$_{eff}$ = 2271 {\\AA}) with different coverage and depth. See Bianchi (2008) for a summary of the UV sources\nclassification and \nstatistics in the main surveys and Morrissey et al. (2007) for instrument description and performance.}\n sky surveys\n(Bianchi 2008, Bianchi et al. 2006, 2007, 2008 and references therein), we have suspected\nthe existence of a substantial number of\n extragalactic objects with FUV-NUV color \nmuch bluer (more negative) than canonical QSO templates and than\nthe majority of QSOs in known samples.\n Such objects are rather ``normal'' at optical wavelengths \n(spectroscopically and photometrically) but they stand out in the observed UV range,\n having FUV-NUV colors similar to those of hot white dwarfs (WD).\nPhotometrically, these objects have UV-to-optical colors similar \nto a stellar binary containing a hot WD and a cooler companion. \nThat a significant number of ``FUV-NUV''-blue extragalactic objects existed \nwas first suspected by Bianchi et al. (2007),\nbased on density counts of photometrically selected WD candidates. In fact,\nthe number of objects per square degree whose SED (FUV to near-IR) is consistent\nwith a single hot WD increases with magnitude down to m$_{UV}$$\\sim$21 (AB)\nand then declines, consistent with Milky Way models. However, the density of\nobjects with similarly blue UV color but redder optical colors, that we would expect to be \n hot WDs with a cool companion, increases considerably \n at fainter magnitudes, suggesting that a significant number of faint extragalactic\nobjects may be included in the color-color {\\it locus} of these stellar binaries\n(Bianchi et al. 2007, Bianchi 2008).\nIn this work we focus on these QSOs, \nwhich display very blue observed FUV-NUV colors, and \n investigate whether their properties are unlike those of known objects.\n\n\n\\section{Sample and data.}\nThe sample was extracted from the catalog of matched UV\/optical sources of Bianchi (2008),\nobtained by matching the UV sources in the GALEX third data release \n(GR3)\\footnote{GR3 is available from the MAST archive at http:\/\/galex.stsci.edu }, to the Sloan Digital Sky Survey (SDSS) \nsixth data release (DR6). \nGALEX provides sky surveys with different sky area coverage and depth: we restricted \nthis work to the ``Medium Imaging Survey'' (MIS) data, which reaches a typical ABmag of $\\approx$22.7 \nin both FUV and NUV. The overlap area between GALEX-GR3 MIS data and SDSS-DR6 is 573 square degrees (Bianchi 2008),\ntaking into account that only the central 1~degree diameter part of the GALEX fields was\nused in our master catalog, to assure homogeneous photometry quality and exclude defects \nin the outer parts of the circular field. \nFor each matched source, GALEX provides FUV (1344 - 1786 \\AA, $\\lambda$$_{eff}$ = 1528 {\\AA})\nand NUV (1771 - 2831 \\AA, $\\lambda$$_{eff}$ = 2271 {\\AA}) photometry, and the SDSS provides {\\it u, g, r, i, z} photometry. \nMore details on the matchings procedure and the catalog are given by Bianchi (2008), Bianchi et al (2009). \n\nIn order to characterize the suspected ``FUV-NUV blue'' QSOs, we \nextracted from the matched UV\/optical source catalog of Bianchi (2008) the\nspectroscopically confirmed QSOs with \n FUV-NUV $<$ 0.1 (AB mag): this FUV-NUV limit\nis ``bluer''(more negative) than the synthetic FUV-NUV color from the two QSO canonical \ntemplates used by Bianchi et al. (2007), which represent average QSO properties,\nat any redshift. The colors of the canonical templates are \nshown in Fig. \\ref{f_ccd} and \\ref{f_ccd2} (cyan diamonds) as a function of redshift.\n We will refer to this sample as ``UV-blue'' QSOs for brevity throghout the paper. \n It is restricted to sources with photometric errors smaller than\n0.3~mag in both FUV and NUV, and color FUV-NUV $<$ 0.1, \nfor which SDSS spectroscopy exists and gives a ``QSO'' classification.\n The requirement of available SDSS spectroscopy effectively limits \nthe sample to brighter magnitudes, but it provides a classification and useful information,\n which will help interpreting larger samples of photometric candidates.\nSuch relatively bright objects will also be accessible to the spectroscopic capabilities of the\nrefurbished HST, and to other follow-up observations. \nNote that the SDSS spectroscopic class ``QSO'' (class 4) probably includes also Seyfert galaxies.\n We will use here the generic term ``QSO'' to\nreflect our selection criterion from the SDSS spectroscopic database. \nIt is important to note that spectroscopic targets in the SDSS were selected with criteria \nunrelated to our present objective and therefore our UV-blue spectroscopically confirmed QSOs may be a \nbiased sub-sample among the UV-blue QSO photometric candidates. \n \n These selection criteria produced an initial sample of 174 objects. \nOne additional object was excluded because its u-band measurement is saturated. \nThe photometric properties of the sample QSOs are presented in \nsection \\ref{s_photo}, and their optical spectra and overall properties\nare analyzed in section \\ref{s_spectra}.\nThe selected objects are shown in two color-color diagrams, Figs.\n\\ref{f_ccd} and \\ref{f_ccd2}, where they can be compared with other classes of objects,\nin particular hot stars, typical QSOs, and galaxies. Our analysis of the spectra (section \\ref{s_spectra})\ngenerally confirms the redshift measurement from the pipeline. However, we found that one object \n(GALEX J172101.08+532433.7, RA=260.2544916, Dec=53.4093516, SDSS match id= 587725490527731868)\nwas misclassified by the SDSS\nspectroscopic pipeline as a QSO with redshift z=2.7: its spectrum is that of a hot star. It is \nshown in some figures because it is interesting to note its position in the color-color diagrams (Figs \\ref{f_ccd} and \\ref{f_ccd2}): the\nGALEX photometry clearly place this object on the stellar sequence and not as a QSO candidate.\nCoordinates, photometry, and other relevant information are given in Table \\ref{t_sample}, and sample images\nare shown in Fig. \\ref{f_fchart_new}.\n\nThe SDSS optical spectra (range $\\sim$ 3800-9200\\AA, resolution $\\sim$1800), provide the initial classification as QSOs and \na measure of redshift for the objects.\nOur sample includes a group of 151 objects at low redshift (0.041$<$z$<$ 0.436), \n and 21 objects with redshift between 1.7 and 2.6, all pointlike.\n Only one object has intermediate redshift (z=0.93) and its identification as a QSO is dubious.\nIt has fairly large photometric errors:\nFUV=21.49$\\pm$0.12, NUV=21.43$\\pm$0.10, but the typical QSO FUV-NUV color at this redshift\nis much redder (by $>$1 mag, see Fig. \\ref{f_ccd}); therefore if it is a QSO it would be quite anomalous. \nAt this redshift the GALEX bands sample rest wavelengths of $\\sim$1300\\AA(NUV) and $\\sim$800\\AA(FUV), and\na very blue FUV-NUV color would be not expected. \nThe image and spectrum of this object are shown in Fig. \\ref{f_oddball} -bottom. \nThe spectrum shows one emission line that is identified as MgII for the alleged redshift and\npossibly a few absorptions including one at the red end that could be H$\\gamma$. \nThe observed wavelengths of the lines are\nnot obvious for identification, assuming other values of redshift the observed line(s) might be\nCIII] $\\lambda$1909 or [OII]$\\lambda$3727, but then other lines such as CIV~$\\lambda$1550 or [OIII]~$\\lambda$5007 should be \npresent and are not.\nSo, either z=0.93 is right or perhaps the emission is some artifact in the spectrum.\nTherefore, we consider this object doubtful and do not include it in our analysis.\nThe lack of objects between redshift 0.5 and 1.7 is consistent with our selection of\nvery blue FUV-NUV color, because Ly$\\alpha$ is in the NUV band between z=0.48 and z=1.63, \ncausing brighter flux in NUV and consequently much redder FUV-NUV color,\nas can be seen in Fig. \\ref{f_ccd}. \n\nWe also caution that while the GALEX FUV and NUV images are taken simultaneously,\nthe SDSS imaging was taken at a different time from the\nGALEX observations, therefore any significant variability may affect the \ncombined UV and optical colors, such as NUV-r. For this reason we based our initial sample\nselection on the FUV-NUV color only. Some of our targets have repeated observations with GALEX,\nbut most repeated measurements are from the AIS (All-sky Imaging Survey), which has about 10 times shorter exposures\nthan MIS (used in this work) and therefore large photometric uncertainties.\nIn a few cases repeated measurements are discrepant by $>$2$\\sigma$ in the combined photometric errors: \nhowever most of the discrepant measurements have artifact flags set. \nWe compile for completeness\n all repeated measurements with exposures longer than 400~sec and formally discrepant by $>$2$\\sigma$\nin Table \\ref{t_dup}, where we provide also comments that help assess reliability, based on the flags \nfrom the pipeline photometry, and our visual inspection of the images. \nWe only excluded measurements on the very edge of the GALEX field (flag ``rim'') however we \ndid not apply error cuts nor area cut, for the purpose of an exhaustive comparison, while our\nanalysis sample is restricted to measurements in the central 0.5~deg. radius of\nthe field for accurate photometry (Sect. 2). \nIn a few cases the discrepancies in the repeated measurements\n cannot be ascribed to artifacts, and these objects may deserve dedicated follow-up photometry. \nIn some cases the variation affects the FUV-NUV color, and in particular some repeated measurements \nhave redder FUV-NUV than our initially selected dataset. \nAll discrepant repeated measurements with MIS exposures (2 high-redshift objects \nand 10 low-redshift objects) have FUV-NUV$>$0.1. If we consider also AIS data \n(exposure times $\\sim$ 100~sec), we find 55 additional \nrepeated measurements discrepant by $>$2~$\\sigma$,\nof which 36 give FUV-NUV redder than our selected dataset (MIS measurements), and other 19 bluer.\n Fast variability is not unknown in QSOs,\nand in particular line strength may vary on short time scales, while it would\nbe less plausible for dust effects to change rapidly. We stress that a variabilty assessment however\nwould require custom photometry, and while the standard pipeline photometry is good for statistical analysis,\nsuch as the scope of this work, \nwe should refrain from overinterpreting individual measurements and individual variations. \nOther 72 AIS and 7 MIS repeated measurements agree within 2$\\sigma$ with our selected measurements given in Table 1. \nOne object in the initial sample, although part of the MIS survey, has a 40~sec exposure \nin FUV and 1518~sec in NUV: although its\nFUV error (0.23) meets our selection limits, a longer MIS exposure of 853~sec in both\n FUV and NUV and smaller errors gives FUV-NUV=0.17, so it \nis eliminated from our analysis sample.\n\n\nA larger sample of about 30,000 QSOs candidates with ``normal'' FUV-NUV colors\n(i.e. similar to the standard template), \nwill be presented elsewhere. We will refer to this sample as ``UV-normal'' \n in the discussion of the UV-blue sample for comparison purposes. \n\n\n\\section{General photometric properties}\n\\label{s_photo}\n\n\nOf the 174 sample objects, 64 sources are classified as point-like (at the resolution of the SDSS \nimaging, $\\sim$1.4$^{\\prime\\prime}$) and 110 as extended (all at low redshift), by the SDSS pipeline.\n We will keep the pipeline classification because it is derived from an objective procedure, although the result depends on\nthe contrast between central source and underlying galaxy. GALEX and SDSS imaging for \na subsample of objects, presented in Fig. \\ref{f_fchart_new},\nshows that the definition of ``pointlike'' (P) or ``extended'' (E) is not clear-cut.\n \n The sample selection, \nas described in section \\ref{s_sample},\nwas restricted to MIS sources with photometric errors less\nthan 0.3~mag in FUV and NUV. Of the 64 point-like\nsources, 42\/33 have errors less than 0.1~mag\/0.05~mag, and only 4\nhave errors between 0.2 and 0.3~mag, in FUV. As for the NUV measurements, \n36\/45 pointlike sources have errors $<$~0.05\/0.1~mag, and 10 have errors\nlarger than 0.2~mag. In the {\\it r-}band, all but 2 objects have errors smaller than\n0.05~mag (one object has an error of 0.14~mag, and one of 0.08~mag).\nOf the 110 extended objects, 108 have {\\it r-}band error $<$0.04~mag,\n97\/94 have NUV \/ FUV error $<$ 0.1~mag. Sources with large photometric errors \nare identified in the figures. \nMost objects with the larger errors are in the z$\\sim$2 group. At this redshift, our color\ncut of FUV-NUV$<$0.1 is more than half a magnitude bluer than the average value for UV-normal QSOs (e.g. Fig.1), \ntherefore these objects may be truly extreme with respect to average samples in spite of their large photometric errors. \nWe point out that the object GALEX J113223.4+641958 (SDSS J113223.42+641958.4) has a u-band magnitude of \n28.7$\\pm$0.46 (petromag) while the magnitude listed on the explorer page for this object is u=25.07$\\pm$3.05.\nIt has no artifact flags, the pipeline records only a warning ``no petrosian radius could be determined. Petrosian magnitude\nstill usable; the object is blended with an extended object''. The surrounding galaxy can be seen in the SDSS\nimaging with a radius of about 5$\\mbox{$^{\\prime\\prime}$}$. We regard the petrosian magnitude as unreliable in the u-band.\nMagnitudes in other bands for this object have smaller errors and seem more consistent among measurements. \nAll other objects have u-band magnitudes brighter than 22.2, consistent with the SDSS limit (see Fig.~3 of Bianchi et al. 2007). \nWe give in Table \\ref{t_sample} petrosian magnitude measurements for the SDSS data, for consistency \namong the sample and with other extragalactic works. \nThe SDSS pipeline also provides magnitudes measured in different ways: psf fitting, \nDeVaucouleurs model, and exponential fitting. A description of the different magnitudes \ncan be found on the SDSSS web site http:\/\/www.sdss.org\/dr5\/algorithms\/photometry.html. \n We checked for all objects whether the different \nmeasurements are discrepant. As expected, for pointlike sources the average difference is \nwithin the 1$\\sigma$ errors and the largest discrepancies close to 3$\\sigma$.\n Disagreement between psf-mag and petromag tend to increase at longer wavelengths, where the\nextended galaxy is contributing. \nFor extended objects, the measurements from petrosian and deVaucouleur-profile fitting agree on average within\nbetter than 2$\\sigma$, while psf magnitudes are more discrepant as expected and should not be used. \n\n The FUV-NUV and NUV-g colors of the sample objects are plotted as a function of redshift in Fig. \\ref{f_uvcolor},\nand the FUV, NUV, and {\\it r-}band magnitudes in Fig. \\ref{f_mags}. \n ``Extended'' sources are plotted with different symbols, to explore possible trends, although the classification\nmust be regarded only as an indication as pointed out above. \nPhotometric errors (1~$\\sigma$ error bars are shown) \nin most cases are quite small compared with the spread in FUV-NUV color\nobserved in our sample. \nFig. \\ref{f_uvcolor} shows that the high-redshift objects have\na wider range of FUV-NUV color than the low-redshift point-like sample, although the spread \nmay simply be caused by the large errors of these faint objects. \nThe lack of extended objects at high redshift is likely due to the fact that for these\nmore distant objects the same imaging does not reveal the underlying galaxy. \nThis question will be investigated with deeper\nimaging aimed at revealing the underlying galaxy in the distant objects and to probe the contrast to the \ncentral source (Hutchings, Scholz, and Bianchi 2009). \n\nFig. \\ref{f_mags} shows that low-redshift pointlike QSOs tend to be brighter than \nextended ones, in both FUV and NUV.\n In the {\\it r-}band, however,\nthe magnitude spread is less (about 4 mags across the sample) and no preferential distribution is \nseen between pointlike and extended samples. Low redshift pointlike objects \nare also brighter (observed magnitudes) than \nhigher redshift objects by about 2-3 magnitudes, but their intrinsic luminosity is lower.\n The distribution of observed magnitudes (left panels) is useful for comparison with other samples,\nand to estimate the possible contamination by these objects to density counts of \nother UV-blue objects such as Milky Way WDs, which have similar FUV-NUV colors (see Bianchi et al. 2007, 2008,\nand Fig. 1), as well as for planning follow-up observations. \nThe misclassified star is shown in these panels. In the right-side panels of Fig. \\ref{f_mags}\nthe absolute magnitudes are plotted (the luminosity distance was derived from the redshift\nusing standard cosmology H$_0$ = 70 km\/s\/Mpc, $\\Omega_M$=0.3, $\\Lambda$ = 0.7 ); \n the distant objects are intrinsically more luminous. \n We plotted for comparison the median absolute magnitudes\nof our UV-normal QSO comparison sample (solid line in the right side plots). \nThe high redshift UV-blue QSOs have luminosities similar to UV-normal QSOs, \nexcept in NUV, where they are fainter.\nThe comparison suggests some absorption in the NUV band (restframe $<$900\\AA~ for z=2) as an\nexplanation of their FUV-NUV color. We will discuss this point later. \n\n \n\\section{Analysis and Discussion.}\n\\label{s_spectra}\nWe discuss the two groups, z$<$0.5, and z$\\sim$2 QSOs, separately \nbecause the FUV and NUV bands sample different restframe spectral regions and therefore\nthe explanations for their blue FUV-NUV colors are different. \n\n\\subsection{The low redshift QSOs}\n\nThe majority of our analysis sample has redshift $<$0.5 (150 objects, 109 extended and 41 pointlike).\n Ly$\\alpha$ transits the GALEX FUV band from z=0.1 to z=0.47, and\nthis fact suggests that an intense \nLy$\\alpha$ emission may be the cause for the ``FUV excess'' of \nthese objects. To test this hypothesis, we constructed templates with \n Ly$\\alpha$ emission enhanced relative to standard templates, \nand derived their synthetic broad-band colors.\n Such {\\it ad hoc} templates\nwith Ly$\\alpha$ enhanced by up to 3$\\times$ match the range of observed FUV-NUV colors of our low-z sample, and\nare shown in Figs. \\ref{f_ccd} and \\ref{f_ccd2} (dark blue diamonds) together with\nsynthetic colors from canonical templates (cyan diamonds), as well as in Fig. \\ref{f_uvcolor}(top).\nNote from these figures that our simple cut of FUV-NUV$<$0.1 produced a low-redshift sample\nbluer in FUV-NUV than standard templates with a spread of about half a magnitude: the redder objects \namong our sample are very close to the standard template\nat z$\\sim$0.2, the UV-bluest objects differ by up to .5~mag (one by $\\sim$1~mag) and\nare concentrated around z$\\sim$0.2 where Ly$\\alpha$ is at the peak of the FUV filter\ntransmission.\nThe modulation with redshift of the hypothetical enhanced-Ly$\\alpha$ effect,\ndue to the filter transmission, is seen in the {\\it ad hoc} template plotted in Fig. \\ref{f_uvcolor}.\n\n\nNone of our sample objects have UV spectra, which would directly reveal the cause of their\nblue FUV-NUV color.\nWe examined their optical spectra and in particular H$\\alpha$, the strongest line in all the objects. \n Fig. \\ref{f_sp_stack_loz} shows the optical spectra\nof our low redshift sample, stacked, and compared with the standard template (cyan). \nThe majority of the pointlike sources have \nemission lines stronger than the average template, and bluer spectral\nslope (flux increasing at shorter wavelengths). For the extended sources, however, line strength is \ngenerally typical and the spectral slope mostly redder than the standard template,\n reflecting the non-negligible contribution by the underlying galaxy (SDSS\nspectra are taken through a 3\\mbox{$^{\\prime\\prime}$} ~ diameter aperture). Sample spectra \n for both pointlike and extended QSOs are also shown in Fig. \\ref{f_sp_loz}.\n There is a wide range of line\nstrengths and profiles, as well as spectral slopes. \n\nThe SDSS pipeline provides automated measurements of width and equivalent width (EW) of the major lines,\nperformed with line fitting; \nwe downloaded and examined those quantities. We found that the centering of the line could be\nused, while line width and equivalent width from the pipeline are not reliable for most\nspectra (examples in Fig. \\ref{f_hawidth}). We remeasured the H$\\alpha$ line, \nfirst by hand to assess the difference from the pipeline measurements,\nand then with an {\\it ad hoc} algorithm for more objective results.\nThe line width estimated by our code is also shown in Fig. \\ref{f_hawidth}. In order to minimize\n the complication of narrow absorptions and emissions in some profiles, \nwe did not measure the width at half maximum (peak) but the width at the average flux value of the \ntotal line emission.\n We consider our measurements more homogeneous than the pipeline values, \nas shown in Fig. \\ref{f_hawidth}, and we use them in the following analysis.\n H$\\alpha$ width, EW and fluxes (F$_{\\lambda}$) measured (at restframe wavelengths) \nfor the low-z sample are reported in Table \\ref{t_meas}. \nErrors from the spectra S\/N and continuum placement uncertainties, are estimated to be less than 10\\%. \nAs a further check, measurements from our code agree with our manual measurements with \nby-eye location of the continuum to a few percent in all but a few cases, where they agree within 10\\%. \nOur measurements include the entirety of the emission feature, no attempt was made to\nseparate narrow components when present, and no correction for [NII] was applied. \n\n We searched for possible correlations of H$\\alpha$ intensity and width, and of the optical spectral slope, \nwith the UV color and absolute luminosity. \nThe spectral slope was measured as the ratio of fluxes integrated in two intervals\nwhich are rather free of conspicuous features in most spectra: \nrest wavelengths 3500-3700\\AA~ and 6000-6400\\AA. \nWe compared several quantities, and we show six interesting cases in Fig. \\ref{f_ha}. \nNo obvious correlation is seen with the FUV-NUV color. \nThe H$\\alpha$ flux, and EW, increase with absolute UV luminosity, \nand to a much lesser extent with u-band luminosity,\nbut not with luminosity at longer wavelengths. \nSimilarly, the optical spectral slope %\npossibly correlates with H$\\alpha$ EW and with UV luminosity\n(but not with optical luminosity: the r-band is also shown in Fig.\\ref{f_ha}):\nit becomes bluer for brighter UV luminosities, suggesting \n that for low QSO luminosity the galaxy relative contribution is more significant. \n There is a clear difference between pointlike and extended sources:\nthe latter have a flatter (redder) optical slope and lower H$\\alpha$ emission, \nreflecting the contribution of the host galaxy. \n This result emphasizes the role of UV studies in extending the known properties of QSOs.\nIf we restrict the sample around redshift z=0.2, \nwhere we have a wider FUV-NUV observed range and Ly$\\alpha$ is at the peak of the filter's transmission,\nthe scatter is much reduced in the correlations with absolute FUV luminosity, and \n some possible correlations with UV color emerge, but the number of points is then\ntoo scarce for robust conclusions. \nAlternative explanations for the blue FUV-NUV color\n may include a dust effect, depressing the NUV flux. However, \nit is not obvious that any known interstellar\nextinction law would have this effect at these redshifts: the 2175\\AA~ dip,\nfor instance, would lie at the upper (long wavelength) end of the NUV passband for redshift beyond 0.2, and the FUV would\nbe more absorbed. There is no correlation of the H$\\alpha$ intensity with foreground E(B-V). \n\n\nFigure \\ref{f_sedLowZ} shows the magnitudes of the low-redshift sample, and their median values\nconnected by lines. The average SED of UV-normal QSOs in this\nredshift range is also shown for comparison. The extended sources differ from the pointlike ones. \nAmong pointlike sources, the overall brightness of UV-normal QSOs is slightly \nlower than the UV-blue QSOs.\nThe extended UV-blue and UV-normal samples have similar SED in the optical bands, showing similarity of the host galaxy\nwhich contributes to the flux. \nWe performed this comparison both using dereddened magnitudes, where each photometric measurement was \ndereddened using E(B-V) (Table 1) estimated from the Schlegel et al. (1998) maps,\nbefore averaging the sample, as well as using observed magnitudes without extinction corrections.\nThe individual sources and the average SEDs shift correspondingly, by up to $\\sim$0.4mag in UV,\nbut the relative differences among average SEDs remain the same. \n\nWhile there is no UV spectroscopy for our objects, we examined UV HST-STIS archival spectra of a\n sample of QSOs %\npublished by Shang et al. (2005). None of them are included \nin the current GALEX MIS coverage, but a few are in the GALEX AIS \n(which has about 10 times shorter exposures than MIS data\nand therefore larger photometric errors).\nWe also computed synthetic colors\nfor the Shang et al sample convolving the observed HST\/STIS spectral fluxes with the GALEX transmission bands.\nWe show their position on the color-color diagram (Fig. \\ref{f_ccd}, small teal diamonds).\nA few of the Shang et al (2005) QSOs have \nFUV-NUV only slightly bluer than 0.1, %\nwhile the rest have typical FUV-NUV colors. \nThe QSOs in the STIS sample with FUV-NUV $<$0.1 have optical spectral slope and H$\\alpha$ emission similar to the standard \nQSO template, but most display stronger Ly$\\alpha$ and CIV, and a range of UV slopes\n(steeper, similar, but also flatter, than the average).\nThe comparison, although limited to a different sample than ours and to a few bright objects, suggests\nthat unusual UV properties may exist that cannot be predicted from the optical data.\n\n\n\\subsection{The high redshift QSOs}\n\\label{s_hiz}\n\nThe photometric SEDs of the 21 UV-blue QSOs with redshift between 1.7 and 2.6 \nare shown in Fig. \\ref{f_jh1}, and their optical spectra in Figs. \\ref{f_sp_hiz} and \\ref{f_sp_stack_hiz}. \nThe QSO with the highest redshift in the sample (z= 2.6)\nhas an extremely red optical spectrum and appears as a very red faint object\nin the optical imaging (Fig. \\ref{f_oddball}).\nIt also has much larger photometric errors than the rest of the sample:\nFUV=22.41$\\pm$0.19 , NUV=22.67$\\pm$0.29, \n{\\it u}=22.22$\\pm$0.8, {\\it g}=21.40$\\pm$0.23, {\\it r}=20.86$\\pm$0.14.\nThe typical QSO FUV-NUV color for \nits redshift is significantly redder (more than half magnitude, Fig. 1)\nso the object may still deserve attention. \n\nAt redshift z=2, the NUV band includes flux in the restframe range\n590 - 940\\AA~ (the filter's $\\lambda$$_{eff}$ becomes restframe 757\\AA) \n and the FUV band includes restframe 450-590\\AA ~($\\lambda$$_{eff}$$\\sim$ restframe 500\\AA).\nWe speculate that a combination of steep flux rise towards restframe extreme ultraviolet (EUV) and absorption below the Lyman \nlimit may explain the observed FUV-NUV color.\nWe constructed spectral templates with FUV flux rising more steeply than in standard templates, using\n two power-law slopes F$_{\\lambda}$ $\\sim$ $\\lambda$$^{\\alpha}$ with $\\alpha$=-0.6 and -1.2.\nThe average slope between 500$--$1200\\AA~ in the large sample of Telfer et al. (2002) is\nF$_{\\nu}$ $\\sim$ $\\nu$$^{-1.76}$, i.e. ${\\alpha}$=-0.24 in F$_{\\lambda}$. \nOur EUV-steep templates are shown with dark green diamonds \nin Figs. \\ref{f_ccd} and \\ref{f_ccd2}. While they have synthetic FUV-NUV color bluer than the canonical template\nat redshift $\\sim$ 2, they are still more than half magnitude redder than the colors\nobserved in our UV-blue sample. The fact suggests that a combination of both EUV flux rise \nat shorter wavelengths and a deep absorption\nbelow the Lyman limit may be required to explain the observed colors of our UV-blue QSOs. \nThe suggestion is supported \nby Fig. \\ref{f_mags}, showing the absolute NUV luminosity of our UV-blue sample to be lower\nthan average. This can also be appreciated in \nFig. \\ref{f_jh1} which shows all the observed magnitudes for our high-redshift sample. The Lyman\nlimit in this redshift range lies between the NUV and u bands, and the\nLyman drop is clearly seen. The line shows the median values, and \n only the object with a red optical spectrum mentioned above differs \nsignificantly (shown by the dotted line). \n Figure \\ref{f_jh1} also shows the median magnitudes for UV-normal QSOs \nfrom the MIS survey, with the same redshift range and error cuts. \nThe average FUV-NUV is much `redder' for the UV-normal QSOs, consistent with our selection. \nThe UV-blue QSOs are fainter in the NUV band, which is sampling\nthe Lyman limit at these redshifts. Thus, our QSOs sample may have somewhat more\nextinction and more severe absorption below the Lyman limit.\n Three of the UV-blue QSOs have strong BAL-type C IV absorption (Fig. \\ref{f_sp_hiz}). \nTheir Lyman discontinuities (estimated from\nthe broad-band photometry, as defined in Fig.\\ref{f_jh2})\nare very large for two of them and smaller than average for one. Thus, it\nis not clear whether BAL absorbers contribute signifcantly to the Lyman drop. \n\n Binette \\& Krongold (2008, and references therein) discuss the spectrum of Ton~34, \nan unusual QSO with %\nan enhanced ``Lyman valley'' in its UV spectra (IUE and HST), which can be reproduced by their models of \nabsorption from carbon crystalline dust (nanodiamonds). Ton~34 is at redshift z=1.93 \nand we investigated whether it could be a possible counterpart of our UV-blue QSOs. \nIt is not included in the GALEX \n surveys to date (it is just outside the edge of an observed GALEX field), so\nwe estimated GALEX FUV and NUV magnitudes by convolving the IUE SW and LW spectra of Ton~34\nwith the GALEX filters, and obtained FUV-NUV $\\sim$1.3,\nclose to the expected color of the UV-normal sample at this redshift and much\nredder than our UV-blue QSOs. This color estimate is uncertain because in the IUE spectra the \nsignal is close to the background limit, and \nHST spectra of Ton~34 do not even cover one of the\nGALEX bands. %\n The GALEX FUV band includes flux longwards of 1344\\AA, \nwhile in the IUE spectrum of Ton~34 the flux is very steeply rising just shortwards of this limit. \nTherefore, a slightly more\nredshifted analog of Ton~34 would produce a much brighter FUV magnitude. \n \nModels of crystalline dust absorption by Binette \\& Krongold (2008) show that the \ngeneral effect is a very deep Lyman valley, and in more detail the relative amounts of absorption in the\nwavelength ranges sampled by the GALEX FUV and NUV filters at z$\\approx$2 vary according to\nthe dust geometry and composition. \nFor example, comparison of dust models in figure A.2 of Binette \\& Krongold (2008) suggests that\na lower column density of the carbon crystalline dust screen (or an intrinsic SED steeper towards short wavelengths)\nmay produce a higher FUV flux, and small grain dust (similar composition as Milky Way\ndust, i.e. silicate and graphite grains, but grain sizes much smaller than MW dust and larger than\nnanodiamonds) would cause a significant depression of the observed-NUV flux but less reduction of the\nobserved-FUV at the redshift of our high-z UV-blue QSOs. \nThis effect would be qualitatively consistent with the SED of our UV-blue QSOs. \nFrom broad-band photometry alone, it is not possible to separate effects of\ndust absorption and intrinsic SED slope, therefore %\nwe can only speculate that the observed FUV-NUV colors in our sample are \nqualitatively compatible with absorption from dust with grains\ndiffering from Milky Way dust (smaller grains), and possibly a steeper flux rise towards EUV. \n The question remains open as to what causes the extremely blue FUV-NUV colors, and whether\nthese objects have known counterparts with similar properties, until UV spectroscopy\ncan be obtained. \n\nWe have measured the emission lines of C IV and C III] (EW, total flux, and full width (FW)\nat 10\\% of the peak flux above the local continuum)\n from the SDSS spectra. Typical errors are of the order of 10\\%.\nLy$\\alpha$ is generally too near the \nend of the optical spectra and could be measured only in ten cases. The C III] line is free of absorptions\nbut some QSOs have significant BAL and interstellar absorptions in the CIV line. \nThe emission line properties do not correlate at all with the FUV-NUV color. \nThe FUV-NUV color does correlate with the g-i color, which may indicate\nthat extinction is involved, and with\nredshift, although weakly, in the sense that higher redshift objects are more\nUV-blue (Fig. \\ref{f_uvcolor} and \\ref{f_jh2}). \nThis is what we would expect in the rest wavelengths below the\nLyman limit, where the continuum is rising again. The Lyman discontinuity \nis larger for higher redshifts too, which is likely caused by where it lies\nbetween the NUV and u bandpasses. \n The emission line EW is larger for fainter FUV magnitudes, but scales more\nslowly than the continuum flux. The line full-width is higher for more luminous\nQSOs, based on their g-band magnitudes (rest frame FUV). \nFlux and EW of C III] and C IV lines correlate with the Lyman discontinuity, but not the line full \nwidth. Thus, there is a connection between line emission and the EUV continuum. \nFigure \\ref{f_jh2} shows some of these correlations; the Spearman's ${\\rho}$\nsignificance test gives a probability of correlation (clockwise from top left)\nof 99\\%, 94\\%, 58\\% and 99\\%. \n\nWhile the dust absorption affects more the continuum, \nthe ionization would be reflected by the line ratios.\n Binette \\& Krongold (2008, and references therein) discuss also \nthe effects of shock ionization versus photoionization. \n It is interesting that their models show low C~IV and N~V relative to Ly$\\alpha$, compared with UV-normal QSOs.\n We measured Ly$\\alpha$$+$NV and C~IV in our high-redshift UV QSOs where possible (10 cases).\nThe line flux ratios may be useful diagnostic since shocks may not be\nrelated to the continuum. %\n Therefore, we also examined SDSS spectra of UV-normal QSOs in the same redshift range \nand compared their line strength with the UV-blue sample. We extracted spectroscopically confirmed\nQSOs in the same redshift range z= 1.7-2.5, but with FUV-NUV $>$0.1, from our master catalog of matched sources. \nWe found 420 objects, compared with our 21 with FUV-NUV $<$0.1. \nWe imposed the same error cuts in FUV, which unfavours\nthe red (normal) QSOs, so the ratio (5\\%) is a lower limit for the\nfraction of UV-blue QSOs compared with normal ones.\nThe relative numbers however may be highly biased because the SDSS spectral targets\nwere chosen with criteria not related to our UV selection. %\nWe measured the same line ratio only for the UV-normal comparison objects with z=2.2-2.5, where Ly$\\alpha$ is included\nin the optical spectra. The measurements are shown in Fig. \\ref{f_ratio}, where a linear\nfit is also shown; the formal probability of correlation is over 99\\%.\n The average line ratios for the UV-blue and UV-normal samples are given in Table \\ref{t_ratios}. \nThe average is 5.2 for our UV-blue sample and 3.7 for our normal comparison sample.\nThe collisional model predicts a ratio of $\\sim$6.7, and the photoionization model 1.8.\nIn one of our UV QSOs the ratio Ly$\\alpha$+NV~\/~CIV is about 10 while Ton~34 has a ratio of 8.7.\nThis bears out the similarity with Ton~34, and a dominance of collisional ionization, compared with \nUV-normal QSOs. \n\n\n\\section{Conclusions and summary}\n We analyzed 171 spectroscopically confirmed QSOs with FUV-NUV color bluer than 0.1, extracted\nfrom the GALEX MIS survey with complemetary SDSS\noptical data. Most of these objects have redshift $<$0.5, and we speculate that \nLy$\\alpha$ emission enhanced up to a factor of 3 with respect to average templates,\nmay explain the observed colors. %\nTheir optical properties are similar to those of UV-normal QSOs. Both photometric and emission line properties differ\nbetween point-like and extended sources, reflecting the contribution from the host galaxy in the latter.\nThe slope of their optical spectra and \nthe strength of H$\\alpha$ (flux and EW) correlate (increase) with intrinsic UV luminosity.\n Ly$\\alpha$ goes through the GALEX FUV band in the\nredshift range of these objects, between 0.1 and 0.5, therefore the resulting effect on \nthe broad-band FUV magnitude is a combination of the line intensity and the filter's transmission curve. \nA restricted sub-sample with redshift around 0.2 (where Ly$\\alpha$ is at the peak of the\nfilter's transmission) seems to show tighter correlations but it is statistically insufficient to support conclusions.\nThe UV luminosity is brighter ($\\sim$0.5 to 1~mag on average) than that of our\nUV-normal comparison sample, the difference being larger in FUV and for the pointlike objects (Fig. \\ref{f_mags} and \\ref{f_sedLowZ}).\n\n\n Our sample of UV-blue QSOs also includes 21 objects with redshift between 1.7 and 2.6. \nTheir photometric errors are generally large, \nthe combined FUV-NUV 1~$\\sigma$ errors are between 0.1 and 0.37~mags, but our FUV-NUV selection limit\n(FUV-NUV $<$0.1)\nis bluer than typical QSO colors at this redshift by more than half magnitude,\nand the observed FUV-NUV colors are bluer than the typical color by up to 1 magnitude or more (Fig. 1).\nFor these UV-blue QSOs at higher redshift we speculate that a combination of unusually strong\nabsorption in GALEX-NUV (restframe $\\sim$600-900\\AA) and EUV-steeply rising flux\n(GALEX FUV $\\sim$ restframe 450-590\\AA) may explain the FUV-NUV color. \nThis is suggested by two facts. First, ad-hoc templates with flux rising towards restframe EUV \nmore steeply than in \n canonical templates, produce observed FUV-NUV colors bluer than the average template \n(Fig. \\ref{f_ccd2}), but still redder than our selection limit by 0.2~mags and redder than\nmost of our UV-blue sample by up to 1 mag. Second, comparison with average SED of \na UV-normal QSO sample, shows the \n NUV luminosity of the UV-blue sample to be fainter, suggesting absorption in the observed NUV.\nDust with composition similar to the typical Milky Way dust but smaller grains \nand carbon crystalline nano-size grains (nanodiamonds)\nwould cause absorption in the observed NUV band, according to the models of \n Binette and Krongold (2008), which may qualitatively account for the observed \nFUV-NUV colors. UV spectroscopy %\nis needed to pinpoint the cause for\nthe FUV-NUV color of these objects. \n The Ly$\\alpha$ to CIV ratio is stronger in the optical\nspectra of the UV-blue QSOs than in the UV-normal comparison sample (at the $>$95\\% confidence\nlevel from K-S test, although both samples are very small, see Fig. \\ref{f_ratio}), suggesting\ncollisional ionization to be more relevant in the UV-blue QSOs.\n\n The group of UV QSOs at z$\\sim$2 may probe a particularly relevant phase\nof galaxy formation, tighly connected with the \nformation of the massive central black hole. In current QSO\/Spheroid coevolution models (e.g. Granato et al 2004)\nthe power of the central QSO rises almost exponentially\nand quickly stops the star formation process.\nDuring the previous phase it is strongly dust enshrouded\nand not visible except in X rays.\nA phase of decreasing \n(but still significant) extinction follows, and finally a shining phase until the fuel is consumed.\nA quick transition is expected between dust-extinguished and\nnot extinguished phases for QSOs at high z.\nThe relative percentage of UV-blue QSOs could be a measure of the relative lifetimes\nof that phase. As explained in section \\ref{s_hiz}, within our sample of matched GALEX-SDSS sources,\nthe number of spectroscopically confirmed QSOs with FUV-NUV$<$0.1 is about 5\\% of those \nwith redder FUV-NUV in the redshift range around 2. However, this number may be highly biased\nbecause the availability of SDSS spectra is serendipitous from the point of view of our selection.\nThe fraction of sources with available spectra is not uniform across the range \nof optical and UV colors, and redshift, of our photometric candidate sample. \nSpectroscopic selection especially favours the brightest samples, while \nBianchi et al. (2007, 2008) show for example a steep increase of UV-blue extragalactic object candidates at\nfaint magnitudes. Some UV-to-optical color ranges are also contaminated by stellar objects and \nthe purity of photometric candidate samples varies greatly according to the colors regime and parameters. \nThe aim of this work was to point out that a non-negligible sample of UV-blue QSOs exist, \nand explore their nature. Statistical considerations will be addressed using a larger sample. \n\n Another possible bias may arise from variability, which is frequently\nobserved in QSOs. Serendipitous repeated UV observations for our sample show \nvariations by $>$3$\\sigma$ in some objects, and in many cases the FUV-NUV in repeated measurements\nis redder than in our selected data-set, making some of these objects UV-normal or close to\nnormal in some of the measurements, and extremely blue in others. We have tried to exclude as\nthoroughly as possible imaging or pipeline artifacts, using the flags provided by the pipeline and\nidentifying several additional unreliable measurements by individual analysis. However, we should\nkeep in mind that pipeline photometry of large datasets has statistical value, and in particular\nthe combination of GALEX and SDSS source catalogs over a large area of the sky proved invaluable to\ncharacterize elusive classes of objects (Bianchi et al. 2007), but it should not be overinterpreted\nfor individual objects.\n\nFor both the low-redshift QSOs where Ly$\\alpha$ may be stronger (possibly up to 3$\\times$) than in\ntypical QSO templates, and the high-redshift QSOs where deep Lyman valley absorption may occur,\nUV spectroscopy is needed for a conclusive explanation of the FUV-NUV color, and to assess\nwhether these are similar to some known objects.\n Our analysis showed that the GALEX photometry provides a unique sieve to select these\nUV-blue QSOs, whose optical properties are not unusual. \n The analysis of this limited spectroscopic\nsample, and its average photometric properties, also provides useful information to separate these QSOs\n from stellar binaries with a hot WD in our larger samples of photometric candidates \n(e.g. Bianchi et al. 2007, 2008), as shown in Figs \\ref{f_ccd} and \\ref{f_ccd2}. \nThe contamination of these objects in stellar samples may\nbe very significant at faint magnitudes, because the density of Milky Way hot WDs, extracted from GALEX catalogs,\nat MIS depth (UV mag $\\sim$ 22.7 ABmag) is much lower than that of QSOs (Bianchi e al. 2007, 2008). \n\n\n\n\\acknowledgments\n\nWe are very grateful to Vahram Chavushyan, Lucio Buson and Sebastien Heinis for discussions\nat different stages of this work, and to the anonymous referee for many comments which\nled to useful clarifications and improved the paper. \nMore information and related papers are available at the author's web site at\n\\url{http:\/\/dolomiti.pha.jhu.edu}. \nGALEX (Galaxy Evolution Explorer) is a NASA Small Explorer, launched in April 2003.\nWe gratefully acknowledge NASA's support for construction, operation,\nand science analysis of the GALEX mission,\ndeveloped in cooperation with the Centre National d'Etudes Spatiales\nof France and the Korean Ministry of \nScience and Technology. \nThe data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). \nSTScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for\n MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts.\n\n\n\n{\\it Facilities:} \\facility{GALEX},\n\\facility{Sloan}, \\facility{HST (STIS)}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\nFrom the introduction of the seat belt to Automatic Emergency Braking (AEB) systems, vehicles have become increasingly safer. Despite the technological improvements, more than 25 thousand fatalities due to road accidents were reported just in the EU in 2017 \\cite{EUraodstat}. These reports also highlighted that $\\approx$\\SI{95}{\\percent} of the fatalities were caused by human errors. Fully autonomous vehicles might present the solution to reduce such accidents by removing the human from the loop. Fully autonomous vehicles, however, may still fail in traffic situation a human could easily handle, such as parking places or pedestrian crossing areas. To smooth the transition from human-driven vehicles to fully autonomous vehicles, teleoperated driving can play a fundamental role.\n\nIn Teleoperated Driving (ToD), a human operator drives the vehicle from a workstation equipped with steering wheel and gas\/brake pedals. The generated control commands are transmitted to the vehicle via a mobile network for execution. The operator can view the vehicle's environment on the monitors as captured by vehicle's cameras. This transmission of signals via the mobile network introduces latency. As Fig.~\\ref{fig:general architecture} shows, there are two types of latency to consider, that are, \\emph{actuator latency} and \\emph{glass latency}. The actuator latency is the time taken by the signals to travel from the input devices at workstation to the actuators on the vehicle via the mobile network. The glass latency is the time taken by camera information to travel from camera on the vehicle to the monitors at the workstation via the mobile network. These latency definitions account for the time spent by signals at various stages in hardware as well as in the network. \n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/Detailed_FTM_System_Architechture_Paper_3_cropped.jpg}\n \\caption{System architecture for Teleoperated Driving showing major components in the closed loop.}\n \\label{fig:general architecture}\n\\end{figure}\n\nToD has several challenges, such as large latency, complete connection loss, or reduced situation awareness. Large latency in the network can lead to unstable real time control of the vehicle \\cite{Pongrac2011GestaltungUE}. To mitigate the effects of latency, techniques like predictive display can be employed. In worst cases, a complete connection loss poses a threat to make teleoperation useless and dangerous at the same time. Nevertheless, strategies like redundant mobile networks are deployed to overcome such an issue \\cite{phantom}. Another challenge in ToD is reduced situational awareness. It becomes difficult to judge velocity and location of the obstacles and this problem aggravates in the peripheral region of the view. In tight driving scenarios, the chances of collision increase and hence a driver assistance system to help avoid collision proves beneficial as shown in \\cite{andi}. As Fig. \\ref{fig:general architecture} shows, the human operator generates steering and velocity commands that are then manipulated by the controller to generate safe collision free control inputs for the vehicle. Since human operator (from the workstation) and the system (with the final authority) control the vehicle, it can be stated as a case of \\emph{uncoupled shared control}~\\cite{reviewSharedControl}. To mitigate the issue of conflict of control interest between human operator and the system, a feedback is necessary to communicate the intent of the system to the human operator.\n\n\n\\subsection{Contributions}\nTo address the above challenges, this paper presents an active safety system for collision avoidance using Model Predictive Control (MPC) in a shared control framework specifically for a teleoperated road vehicle, building on the approach presented in \\cite{andi}. First, we propose a driver assistance system (Sec. \\ref{sec: section2}), based on MPC, for a teleoperated semi-autonomous road vehicle. The primary objective is to track the input references as generated by the human operator and intervene when deemed necessary to avoid collision with static or dynamic obstacles. As compared to \\cite{andi}, the approach presented in this paper controls both, the longitudinal and lateral dynamics of the vehicle. In addition, our design introduces a restriction on the controller's authority to override human operator to improve the operator trust on the automation. Furthermore, we introduce potential fields in the MPC formulation to efficiently achieve a collision free manoeuvre. Second, we propose a visual feedback (Sec. \\ref{sec: section3}) to communicate system's intent and enhance trust in the human operator over the system to mitigate conflict of control interest. Third, we propose an MPC data-based predictive display technique (Sec. \\ref{sec:predictive}) to mitigate effects of high latency in a teleoperation setup. Finally, the proposed approach is validated using a high-fidelity vehicle simulator with the simulated human operator (Sec. \\ref{sec: section5}).\n\n\n\\subsection{Related Work}\\label{sec:RelatedWork}\nOur work spans across different areas. For this reason, this section provides an overview of the relevant literature on shared control, obstacle avoidance, visual feedback and predictive display with a main focus on MPC-based approaches.\n\n\n\\paragraph*{Shared Control}Several shared control approaches have been proposed in the literature, such as model-based coupled shared control, model-based uncoupled shared control, model-free uncoupled shared control (as defined in~\\cite{reviewSharedControl}). In uncoupled shared control, automation has final authority to implement the input to the vehicle unlike in coupled shared control. A model-based shared control models the shared control behaviour, mostly using a driver model. A model-based coupled shared control approach is followed in \\cite{Ercan2016TorquebasedSA}, where human driver and steering system models are incorporated into a lateral controller design. Modelling the driver requires actual driving tests as shown in \\cite{pred_setinvariance} and the driver model becomes specific to a particular driving style.\n\nA model-free uncoupled shared control approach is followed in ~\\cite{anderson}, where a final steering command as a weighted combination of driver and automation input according to a tailored threat metric is computed. Tuning of this threat metric is not trivial and only lateral control was showcased in this work.\nFinally, a model-based uncoupled shared control approach is followed in \\cite{safeNL}, \\cite{sharedcontrol} and \\cite{andi}. In \\cite{safeNL}, a model predictive contouring control (MPCC) framework is used for automation where center of the lane is considered as a reference path. Another optimization problem considers driver inputs and a decaying function is used to compute a weighted combination of driver and automation inputs. This approach considers both lateral and longitudinal control, but cannot be used for the case of teleoperation where availability of information of lane center via perception module is not reliable. The authors in \\cite{sharedcontrol} and \\cite{andi} directly incorporate driver's steer commands into the objective function of MPC for only lateral control, where the objective is to match the driver inputs.\n\n\\paragraph*{Obstacle Avoidance} Obstacles and ego vehicle can be directly modelled in the constraints of the MPC formulation as \\emph{geometrical shapes}, based on which a collision free criterion can be incorporated into the problem formulation of controller, as proposed, for example, in \\cite{safeVRU} and \\cite{MPClaura} in the context of fully autonomous vehicles. Reactive methods, such as potential fields, can also be incorporated in the MPC formulation instead of explicitly modelling obstacles in the constraints. The authors of~\\cite{PF} represented crossable and non-crossable obstacles with different potential fields. The authors of~\\cite{andi} modelled the obstacles as higher-order ellipses and formulated a potential function such that the front vehicle corners are pushed away from the obstacles. This potential function is incorporated into the objective of the MPC problem formulation. The advantage of using potential function over spatial constraint is that the size of the optimization problem does not grow with the number of obstacles in the scenario \\cite{safeNL,safeVRU}. Therefore, the solution time of the problem is not expected to increase with the increase in the number of obstacles. \n\n\\paragraph*{Visual Feedback} A feedback is necessary for the controller to communicate its intent to the human driver. A haptic feedback on steering wheel is used in \\cite{hapticFTM}, to assist human operator in challenging situations and helps prevent collision of teleoperated vehicle. However, haptic feedback on steering wheel can be used only for lateral control. The authors in \\cite{VREUGDENHIL2019371} showcased the benefit of having visual feedback along with haptic shared control for obstacle avoidance. Haptic feedback on steering wheel cannot communicate future vehicle state that might affect current driver inputs. This is supposedly compensated by a visual feedback. Monitors in teleoperation can be used to illustrate future vehicle positions. This advantage is exploited in this work.\n\n\\paragraph*{Predictive Display} As presented in \\cite{ftmlatency}, predictive display can be used to predict ego vehicle position, to compensate for the delay in signal transmission.\nThis involves projecting the predicted vehicle position images on the video stream. However, these techniques assume constant vehicle velocity and road wheel angle in the predictions.\n\n\n\\section{SHARED CONTROL APPROACH}\\label{sec: section2}\n\n\\subsection{Prediction Model}\nThe teleoperated vehicle is expected to be driven at slow speeds with moderate driving. Hence, the prediction model chosen in this approach for the MPC is the following kinematic bicycle model~\\cite{Borrelli_kinematic}:\n\\begin{comment}\n{Therefore, a kinematic bicycle model can help predict the motion of the vehicle. As given in \\cite{Borrelli_kinematic}, the following equations describe the vehicle model equations in the inertial frame for the center of mass (CoM):}\n\\end{comment}\n\\begin{subequations}\\label{eq:kinematic}\n\\begin{align}\n \\dot{x} &= v\\cos{(\\psi + \\beta)} \\label{eq:kinematic1}\\\\ \n \\dot{y} &= v\\sin{(\\psi + \\beta)} \\label{eq:kinematic2}\\\\ \n \\dot{\\psi} &= \\frac{v}{l_r}\\sin{(\\beta)} \\label{eq:kinematic3}\\\\ \n \\dot{v} &= a \\label{eq:kinematic4}\\\\\n \\beta &= \\tan^{-1}{\\bigg(\\frac{l_r}{l_f + l_r} \\tan(\\delta)\\bigg)} \\label{eq:kinematic4}\n\\end{align}\n\\end{subequations}\nwhere $x$ and $y$ are coordinates of the center of mass (CoM) of the vehicle, $\\psi$ is the heading angle of the vehicle in the inertial frame, $v$ is the vehicle velocity, $l_f$ and $l_r$ are the distances between CoM and front and rear axles respectively. $\\beta$ is the angle that the velocity vector makes with the longitudinal axis of the vehicle, $a$ is the acceleration of the CoM and $\\delta$ is the road wheel angle (RWA) of the front wheels. To simplify the notation, we define the following states and input quantities, that are, $z := [x,y,\\psi, \\delta, v]$, and $u := [\\dot{\\delta}, a]$ respectively.\n\n\\subsection{Modelling Ego Vehicle and Rectangular Obstacles}\nThe collision avoidance criteria are incorporated in the MPC problem formulation by approximating the ego vehicle and the rectangular obstacle as geometric shapes. The ego vehicle is represented with four longitudinally shifted circles of radius $r$, along its longitudinal axis ~\\cite{MPClaura}. The benefit of representing the ego vehicle with circles instead of just using front vehicle corners as in \\cite{andi} for collision avoidance is that it ensures that the ego vehicle stays collision free from the obstacles with a downside of an increase in the computational burden. A rectangular obstacle is represented with a higher order ellipse. The implicit function, as proposed in \\cite{andi}, that describes the shape of the obstacle in inertial frame is given by,\n\\begin{equation}\\label{eq:ellipse}\n\\resizebox{0.9\\hsize}{!}{$\n e_{i}^m(x_{c_{i}},y_{c_{i}}) = \\bigg(\\frac{R(\\phi)(x_{c_{i}} - x_{\\textrm{obs}}^m(t))}{\\alpha_{\\textrm{maj}}}\\bigg)^n + \\bigg(\\frac{R(\\phi)(y_{c_{i}} - y_{\\textrm{obs}}^m(t))}{\\beta_{\\textrm{min}}}\\bigg)^n - 1\n $}\n\\end{equation}\nfor $i$ $\\in$ $\\{1,2,3,4\\}$. Here, $(x_{c_{i}}, y_{c_{i}})$ represents the center of the $i^{th}$ ego vehicle circle and $(x_{\\textrm{obs}}^m(t), y_{\\textrm{obs}}^m(t))$ represents the center of the $m^{th}$ obstacle. $R(\\phi)$ is the rotation matrix, depending on the heading $\\phi$ of the obstacle. The semi-axis of the $\\alpha$ and $\\beta$ and the even order $n$ of the ellipse help define the size of the bounding ellipse. Therefore, we define \n\\begin{equation}\n \\alpha_{\\textrm{maj}} = a + r \\quad \\text{and} \\quad \\beta_{\\textrm{min}} = b + r, \n\\end{equation}\nwhere $a := f \\cdot L\/2,$ $b = f \\cdot B\/2$, $f = \\sqrt[n]{2}$ and $r$ is the radius of the ego vehicle. $L$ and $B$ are the length and breadth of the obstacle respectively.\n\n\\subsection{Potential Field for Collision Avoidance}\\label{subsec: PF}\nBuilding on \\cite{andi}, we rely on a repulsive potential field approach for collision avoidance. The potential field $P$ is a sum of individual potential functions for $m$ obstacles for all the $i$ circles of the ego vehicle, that is,\n\\begin{equation}\n\\label{eq:potential_function_all}\n P = \\sum_{m} \\sum_{i} P_{i}^m (x,y),\n\\end{equation}\nwhere the potential function, for the $i^{th}$ circle for the $m^{th}$ obstacle is as follows\n\\begin{gather}\\label{eq:pot_function}\n P_{i}^m (x,y) = \\frac{\\tau}{(e_{i}^m(x_{c_{i}},y_{c_{i}})+1)^{\\rho}}.\n\\end{gather}\nIn the equation above, $\\tau$ and $\\rho$ are the design parameters that specifies the strength and slope of the potential function. It should be noted that $e^m$ is the implicit function that represents the shape of the $m^{th}$ obstacle, which is already given by \\eqref{eq:ellipse}. When the ego vehicle is in collision with the obstacle, the value of $e^m$ becomes zero and hence $P_{i}^m$ equates to $\\tau$. To make the obstacle as non-crossable for the vehicle, this information is used to formulate a constraint in MPC formulation.\n\nThe potential field approach for collision avoidance ensures that the size of MPC problem stays constant irrespective of the number of obstacle, an advantage over the spatial constraint approach.\n\n\\subsection{Model Predictive Control Formulation}\nWe rely on MPC to achieve the navigation objectives. The MPC controller repeatedly solves a constrained optimization problem to compute an optimal sequence of control commands that minimize a desired cost function $J$ over a finite horizon $N$. At every sampling instance, only the first element of this control sequence is applied in closed loop. This section details our MPC problem formulation. \n\nThe objectives of the controller are \\emph{(i)} to match the human operator's inputs (i.e., $\\delta$ and $v$), \\emph{(ii)} ensure collision avoidance, and \\emph{(iii)} restrict the controller authority to intervene to avoid surprises for the human operator. To achieve the objectives above we define the following MPC:\n\\begin{subequations}\n\\begin{align}\n\\underset{z, u, S}{\\min} &\n \\sum_{k=1}^N J_k(z_k,u_k, S_k) \\label{eq:mpc_a}\n\\end{align}\n\\begin{align}\n\\text{subject to} \\quad & z_{k+1} = f(z_k, u_k) \\label{eq:mpc_b}\\\\\n& z_0 = z(t) \\label{eq:mpc_d}\\\\\n& \\delta_{\\textrm{min}} \\leq \\delta_{k+1} \\leq \\delta_{\\textrm{max}} \\label{eq:mpc_e}\\\\\n& v_{\\textrm{min}} \\leq v_{k+1} \\leq v_{\\textrm{max}} \\label{eq:mpc_f}\\\\\n& \\Dot{\\delta}_{\\textrm{min}} \\leq \\dot{\\delta}_k \\leq \\Dot{\\delta}_{\\textrm{max}}\\label{eq:mpc_g}\\\\\n& a_{\\textrm {min}} \\leq a_k \\leq a_{\\textrm{max}} \\label{eq:mpc_h}\\\\\n& \\resizebox{0.65\\hsize}{!}{$\n\\delta_{\\textrm{dev min}} - s^{\\delta}_{k+1} \\leq \\delta_{k+1} - \\delta_{\\textrm{ref}}(t) \\leq \\delta_{\\textrm{dev max}} + s^{\\delta}_{k+1}\n$}\\label{eq:mpc_i}\\\\\n& P_{k+1,i} \\leq \\tau + s^{p}_{k+1} \\label{eq:mpc_j}\\\\\n& k\\in (0,N-1) \\notag, \n\\end{align}\n\\end{subequations}\nwhere $z_k$, $u_k$, and $S_k$ are the state, command, and slack variables (discussed below) at prediction step $k$. The MPC controller minimizes the cost \\eqref{eq:mpc_a} subject to the constraints~\\eqref{eq:mpc_b}-\\eqref{eq:mpc_j}. The cost $J_k$ is defined as follow:\n\\begin{equation}\n J_k:= J^{P}_k + J^{\\delta}_k +J^{v}_k+ J^{S}_k.\n\\end{equation}\nThe first term of the cost function, namely $J^{P}$, is associated with the potential field to ensure a collision free motion and it is defined as follows: \n\\begin{equation}\n J^{P}_k = W_{P} P_{k}(x_k,y_k),\n\\end{equation}\nwhere $W_P$ is a weighting factor and $P$ is defined according to~\\eqref{eq:potential_function_all}, for all the circles representing the ego vehicles and for all the obstacles. \n\n\\noindent The second term of the cost function penalizes the deviation of controller input $\\delta$ from the reference or human operator's input $\\delta_{\\textrm{ref}}$ with the penalty $W_{\\delta}$,\n\\begin{equation}\n J^{\\delta}_k = W_{\\delta} (\\delta_{\\textrm{ref}}(t) - \\delta_{k})^2\n\\end{equation}\nThe third term of the cost function penalizes the deviation of controller input $v$ from the reference or human operator's input $v_{\\textrm{ref}}$ with the penalty $W_{v}$\n\\begin{equation}\n J^{v}_k = W_{v} (v_{\\textrm{ref}}(t) - v_{k})^2\n\\end{equation}\nThe last term in the cost function is used to penalize, with a penalty $W_{s}$, slack variables $S_k:=[s^{{\\delta}}_{k},\\,s^{p}_{k}]^{\\textrm{T}}$ which are used to soften constraints~\\eqref{eq:mpc_i} and~\\eqref{eq:mpc_j}:\n\\begin{equation}\n J^{S}_k = W_{s} S_{k}^2.\n\\end{equation}\nConstraint \\eqref{eq:mpc_b} represents the dynamic coupling. In addition, note that $f(z_k, u_k)$ represents the discritized kinematic bicycle model equations presented in equations \\eqref{eq:kinematic}. \nConstraints \\eqref{eq:mpc_e} and \\eqref{eq:mpc_g} limits the RWA $\\delta$ and its rate of change $\\dot \\delta$, respectively. Furthermore, Constraints \\eqref{eq:mpc_f} and \\eqref{eq:mpc_h} limit the velocity $v$ and acceleration $a$ of the vehicle, respectively. \nConstraint \\eqref{eq:mpc_i} restrict the controller's authority to deviate from the reference RWA at time $t$. This makes the controller action moderate in the lateral sense and helps avoid larger mismatch between human operator's intentions and the controller's action.\nFinally, together with the potential field in the cost function, constraint \\eqref{eq:mpc_j} helps ensure collision avoidance by making the obstacles uncrossable. The upper limit of this constraint is governed by $\\tau$ as found in \\eqref{eq:pot_function}. Constraint \\eqref{eq:mpc_j} is softened to practically avoid infeasible solutions, while a high penalty on the slack variables in the cost function limits such a constraint violation.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale= 0.21]{images\/RvizPlot.png}\n \\caption{Rviz\\protect\\footnotemark illustration of visual feedback to be added on top of video stream on the screens at workstation. The maximum and minimum vehicle trajectory due to the corresponding controller intervention is shown by the green\n controller authority cone. Obstacles are represented by red cubes.}\n \\label{fig:ros}\n\\end{figure}\n\\footnotetext{http:\/\/wiki.ros.org\/rviz}\n\n\\section{MPC DATA-BASED VISUAL FEEDBACK}\\label{sec: section3}\nTo further raise human operator's trust and understanding of the system behaviour, a feedback can be used to communicate the intention of system to the human operator as discussed in \\ref{sec:RelatedWork}. Our proposed visual feedback contains information on the region of future vehicle positions due to controller authority, as defined by constraint \\eqref{eq:mpc_i}, and future vehicle path, shown by white tracks, due to complete sequence of steering wheel angle and velocity inputs by controller. This feedback is proposed to be added as graphical illustration on top of the video stream on screens at workstation. The visual feedback is expected to help the human operator better understand the possible positions of the vehicle according to both the operator's and controller's inputs. It aims at avoiding any surprises or conflict if the controller intervenes. This concept is illustrated in Fig. \\ref{fig:ros}.\n\n\\section{MPC DATA-BASED PREDICTIVE DISPLAY}\\label{sec:predictive}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/MPC_Predictive_Display_Architechture_Paper_3_cropped.jpg}\n \\caption{System Architecture for MPC Data-Based Predictive Display.}\n \\label{fig:pred arch}\n\\end{figure}\nThis section discusses our proposed predictive display method. Our predictive display exploits the MPC predictions to mitigate the effects a high network latency can have on the teleoperation. Recall that the reference module in Fig. \\ref{fig:general architecture} generates reference inputs based on the state feedback it receives, which is delayed by time equal to glass latency. These reference inputs are further delayed by a time equal to the actuator latency during the transmission to the MPC module. During this time the ego vehicle would have travelled by time equal to this total round trip latency. The goal of our predictive display is to attempt predicting the ego vehicle position during the time frame of a total round trip latency. This predicted vehicle position can be used to enrich the camera images before they are displayed on the screens at the workstation. \n\nAuthors in \\cite{ftmlatency} propose a \\emph{full prediction} method, which relies on a bicycle model (as presented in \\eqref{eq:kinematic}) for prediction assuming constant vehicle velocity and RWA. This method is taken as baseline for comparison with the proposed MPC data based Predictive Display in section \\ref{sec:predictive results}.\n\n\nIn our approach, the idea is that MPC's predicted state corresponding to time equal to total round trip latency could be used as state feedback.\nThis simplifies the process and needs no new calculations as in \\cite{ftmlatency}. MPC already uses a model of the system and sets of constraints and objective to calculate optimal evolution of states and inputs. This method also eliminates the assumptions of constant velocity and RWA. Fig. \\ref{fig:pred arch} illustrates this concept.\n\n\n\\section{SIMULATION RESULTS}\\label{sec: section5}\nThe validation of the approach was carried out with simulations in MATLAB\/Simulink\\textsuperscript{\\textregistered}. For simulations, a high fidelity car model from IPG CarMaker~\\cite{cm} was used as plant to test the designed controller. A Volvo XC90 T6 AWD was used as the testing vehicle. The optimization of MPC problem is carried out through \\texttt{acados}, an open source software package with a collection of solvers for fast embedded optimization intended for fast embedded applications \\cite{acados}. Sequential Quadratic Programming (SQP) is used as the solver by \\texttt{acados}. The algorithm runs well below \\SI{50}{ms} on an Intel Core i7-4600 2.10 Ghz 2 Core CPU with 8 GB of RAM and hence it can be expected to perform even better when run on a powerful hardware of the experimental vehicle. The system architecture for ToD is shown in Fig. \\ref{fig:general architecture}. The architecture consists of two main blocks, that are, \\emph{(i)} the workstation, where the human operator generates reference inputs of velocity and RWA, \\emph{(ii)} the teleoperated vehicle equipped with the proposed MPC controller. In a real setup, the signals between these two blocks are transmitted over a commercial 4G LTE network and hence delayed due to network latency.\nIn a simulation setup, delay blocks are added separately to artificially introduce this latency in simulations.\nFor simulations, the reference block generates reference velocity, whose values are predefined as a function of time. In addition to this, the reference block simulates the human operator using a Feedback Linearized Path Tracking Controller (FBLC), taken from \\cite{fblc1}. The control law used to simulate the human operator is defined as follows\n\\begin{equation}\n \\delta_{\\textrm{FBL}}(t) = \\arctan\\Big(\\frac{-\\gamma_{1} e_{L}(t) - \\gamma_{2} v \\sin{(e_{H}(t))}}{ v^2 \\cos(e_{H}(t))}\\Big)\n\\end{equation}\nwhere, the lateral and heading errors are $e_{L}$ and $e_{H}$, respectively, and $v$ is the current vehicle velocity. The correction for heading error and lateral error can be prioritized using the gains $\\gamma_{1}$ and $\\gamma_{2}$. The errors, $e_{L}$ and $e_{H}$ are computed with respect to the reference tracking point along the trajectory, which is again a function of lookahead distance. To include the effect of visual feedback to the human operator, we added another feedback term. This term penalizes the deviation between the operator SWA (RWA) or the calculated $\\delta_{\\textrm{FBL}}$ and the current SWA (RWA), with the proportional gain $\\gamma_{3}$. Thus, the overall simulated human operator's RWA input becomes:\n\\begin{equation}\n \\delta_{\\textrm{ref}}(t) = \\delta_{\\textrm{FBL}}(t) + \\gamma_3(\\delta(t_{0}) - \\delta_{\\textrm{FBL}}(t))\n\\end{equation}\n\nThe CarMaker plant cannot take velocity input directly and hence a Cruise Control (CC) is designed to translate the desired velocity from MPC into corresponding Gas and Brake pedal values. A basic CC is formulated based on a proportional-integral controller. \n\nThe sampling time of MPC is set to $t_s=$ \\SI{50}{ms} since the solution times are well within this limit. The order of the ellipse $n$ is taken as \\si{4}. The potential function parameters are tuned to $\\tau=$ \\num{0.1} and $\\rho=$ \\num{2}. This tuning results in a close manoeuvre of the vehicle around the obstacles. \nToD is expected to be performed at low speed due to the perils of remote driving at high speed. Therefore, these simulations consider only low speed (\\SI{3}{m\/s}) scenarios where the effect of latency is not so crucial and hence a constant latency in the simulation is considered. In the following, the performance of the proposed controller, namely, our Active Safety System (ASS), is compared with a baseline approach which does not have longitudinal control capability and restriction on deviation of RWA from its corresponding reference. Section \\ref{sec: overtake} compares the two approaches in a scenario that involves overtaking a stationary obstacle with another dynamic obstacle in the scene. In addition, Section \\ref{sec:predictive results} illustrates our MPC-data-based predictive display concept.\n\n\\subsection{Overtake Scenario}\\label{sec: overtake}\n\\begin{figure}[]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/compOvertake\/paperPFOvertkDCYL_fig1-eps-converted-to.pdf}\n \\caption{Trajectory Driven for Overtake Scenario. Progression of ego vehicle with ASS is shown in shades of blue. Progression of ego vehicle with Baseline controller is shown in shades of green. Progression of oncoming obstacle (dynamic) is shown in shades of red. Stationary obstacle is shown in grey color. The shades go from light to dark with the progression of time.}\n \\label{fig:overtake fig1}\n\\end{figure}\nThis scenario considers the ego vehicle trying to overtake a stationary obstacle along its path (e.g., a parked car). While overtaking, the ego vehicle has to deal with an upcoming vehicle proceeding in the opposite lane.\nAt every sampling time, the dynamic obstacles are elongated in size over the prediction horizon assuming their current heading and velocity stays constant. Effectively converting them into large static obstacles with changing locations and dimensions. Thus, collision avoidance of the dynamic obstacles is accounted for in a conservative manner, with a possibility of reducing this conservatism by accounting for predicted obstacle positions along the horizon by a perception module.\nFig. \\ref{fig:overtake fig1} shows the behavior of the ego vehicle with our ASS approach. The reference path (green dashed line) of the ego vehicle's CoM is purposefully designed in such a way that the ego vehicle collides with the stationary obstacle while going around. The ASS-based ego vehicle brakes when encountering the oncoming obstacle, and then overtakes the stationary obstacle when there is free space. This behavior is reflected in the velocity and RWA results in Fig.~\\ref{fig:overtake v and delta}, which compares our method (blue lines) with the baseline controller (red lines). The velocity plot in Fig. \\ref{fig:overtake v} highlights the contribution of our ASS on the control of the vehicle. As the figure shows, our system reduces the vehicle velocity to prevent collisions, in contrast with the constant reference signal provided by the operator. As Fig. \\ref{fig:overtake delta} highlights, our ASS avoids collision by correcting the RWA at time \\SI{40}{s}, where the RWA differs from reference RWA. Thus, longitudinal and lateral control capabilities of the ASS are highlighted here. The vehicle with Baseline controller fails to stop after encountering the oncoming obstacle and collides with it since it does not have any longitudinal control capability. The end of its trajectory represents the point where the vehicle leaves the road boundary and hence the simulation gets aborted by CarMaker. This highlights the drawback of not having longitudinal control capability. In addition to this, the RWA plot in Fig. \\ref{fig:overtake delta} also shows a large deviation of RWA from reference RWA. This steering behaviour is highly counter-intuitive to what the human operator wishes to do, potentially raising trust issues over the system. In comparison to this, even though the ASS steering profile is showing considerable deviation from the reference RWA, it remains well around the range of reference and follows the reference back again. This highlights the importance of restricting the controller's authority to deviate from the reference RWA by using the constraint \\eqref{eq:mpc_i}.\n\n\n\\begin{figure}[t]\n \\captionsetup{justification=centering}\n \\centering\n \\begin{subfigure}[b]{\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/compOvertake\/paperPFOvertkDCYL_v_small-eps-converted-to.pdf}\n \\caption{Velocity}\n \\label{fig:overtake v}\n \\end{subfigure}\n \\newline\n \\begin{subfigure}[b]{\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/compOvertake\/paperPFOvertkDCYL_delta2_small-eps-converted-to.pdf}\n \\caption{RWA $\\delta$}\n \\label{fig:overtake delta}\n \\end{subfigure}\n \\caption{Longitudinal velocity and RWA of the ego vehicle in the overtake scenario.}\n \\label{fig:overtake v and delta}\n\\end{figure}\n\n\\subsection{Predictive Display}\\label{sec:predictive results}\nThe predictive display is expected to nullify the effect of latency and produce results equivalent to that for no latency in system. Fig. \\ref{fig:predictive fig1} compares the performance of our ASS with our proposed predictive display (dashed purple line) with the baseline approach~ \\cite{ftmlatency} (solid light blue line) and our ASS without predictive display (solid red line). The performance of our method with no latency is used as a benchmark. In addition, the stationary obstacles are represented by higher order ellipse (brown) and they are inflated by the radius of the ego circle as represented by the large ellipses (red). The reference trajectory (green dashed line) is purposefully designed for a collision with obstacles. In this simulation, we consider a total round trip latency of \\SI{500}{ms}.\nThe trajectory driven by the CoM of the ego vehicle for both the cases with predictive displays (Baseline and MPC data-based) overlap the trajectory with no latency in the system. This proves that both predictive display techniques equally nullify the effect of latency. Nevertheless, the advantage of the proposed MPC-data-based predictive display method over the baseline method lies in the fact that it does not assume constant velocity and RWA for predicting vehicle position. It also removes extra calculation step of predicting vehicle position using a bicycle model as done in \\cite{ftmlatency}. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/PFpredictiveCompare\/paperPFcompPredDispSCYL_fig1-eps-converted-to.pdf}\n \\caption{Trajectory of the CoM of the ego vehicle driving through closely placed obstacles with latency.}\n \\label{fig:predictive fig1}\n\\end{figure}\n\nIt should be noted that, even in the absence of any latency mitigation technique, the vehicle trajectory stays collision free, verifying that the controller is efficient to avoid collision even with delayed (outdated) reference inputs.\n\n\\section{CONCLUSION}\nThis paper presented an ASS approach for teleoperated vehicles based on model predictive control. The ASS controls both the lateral and longitudinal vehicle's dynamics allowing the vehicle to perform a large range of maneuvers and avoid collision by both braking and steering intervention. Collision avoidance is achieved by using artificial potential fields in the MPC controller. Our ASS also has an added restriction on the controller's authority to prevent large deviations from the human operator's RWA inputs. In addition, we proposed a visual feedback in the form of graphical presentation on the monitors of workstation. \nFinally, to mitigate the effects of large latency in a teleoperation setup, we proposed a novel MPC data-based predictive display method. We tested our approach by using a high fidelity vehicle model for plant and a realistic environment.\nThe simulation results highlighted the advantages of our ASS for teleoperation compared to existing baselines. Future work includes experimental validation of the proposed work.\n\n\n \n \n \n \n \n\n\n\n\n\n\n\n\\newpage\n\\section*{APPENDIX}\n\nThe MPC penalties and FBLC gains for the corresponding scenarios are summarized in the following table. \n\n\\begin{table}[h!]\n\\centering\n\\caption{MPC penalties and FBLC gains.}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{1}{l|}{$W_{P}$} & \\multicolumn{1}{l|}{$W_{\\delta}$} & \\multicolumn{1}{l|}{$W_{v}$} & \\multicolumn{1}{l|}{$W_{s}$} & \\multicolumn{1}{l|}{$\\gamma_1$} & \\multicolumn{1}{l|}{$\\gamma_2$} & $\\gamma_3$ \\\\ \\hline\nOvertake Scenario & 0.1 & $10^3$ & 1 & $10^5$ & 1 & 2 & 0.25 \\\\ \\hline\nPredictive Display & 0.1 & $10^2$ & 1 & $10^5$ & 1 & 2 & 0.25 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section*{ACKNOWLEDGMENT}\nThis work was presented at the workshop for Road Vehicle Teleoperation (WS09), IV2021.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}