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+{"text":"\\section{Introduction}\n\nMrk~766 is a bright ($F_{(2-10)}\\sim 2 \\times 10^{-11} \\rm erg\\,\ncm^{-2} s^{-1}$), soft ($\\Gamma_{0.1-2.4} \\sim 2.7$) X-ray source at\nredshift $z=0.012$.  The spectrum measured with the {\\it Einstein} IPC\nand MPC was complex and ultra-soft ($\\Gamma=1.77$; $kT=18.6 \\rm eV$;\nUrry {\\it et al.}\n\\markcite{34} 1990).  A shortest time scale of variability of 1000\nseconds and a steep and variable power law index was found in a long\nobservation using {\\it EXOSAT} (Molendi, Maccacaro \\& Schaeidt\n\\markcite{20} 1993).  During the {\\it ROSAT} All Sky Survey, \nMrk~766 was bright ($F_{0.1-2.4} \\sim 1.5\n\\times 10^{-10} \\rm erg\\, cm^{-2} s^{-1}$ (unabsorbed)) and\nvariability by a factor of three with no accompanying spectral\nvariability was observed in 10--12 hours (Molendi, Maccacaro \\&\nSchaeidt \\markcite{20} 1993).  Pointed {\\it ROSAT} observations\nrevealed spectral variability that Netzer, Turner\n\\& George (1994) \\markcite{24} showed could not be explained by a\nchange in ionization of a warm absorber, and Molendi \\& Maccacaro\n(1994) \\markcite{19} attributed to a change in the accretion rate.\n\nMrk~766 is a member of the X-ray narrow line Seyfert 1 (NLS1) galaxy\nclass (Osterbrock \\& Pogge \\markcite{25} 1985; Goodrich \\markcite{13}\n1989).  {\\it ROSAT} observations of NLS1s find soft 0.1--2.4 keV X-ray\nspectra and rapid, large amplitude soft X-ray variability.  The soft\nX-ray spectra of NLS1s are systematically steeper than the spectra of\nbroad-line Seyfert 1 galaxies (Boller, Brandt \\& Fink \\markcite{3}\n1996 and references therein). A harder high energy power law component\ngenerally was not observed in the relatively soft {\\it ROSAT} band.\nOnly a few observations at higher energies have been reported.  The\n{\\it ASCA} spectrum of the NLS1 object IRAS~13224-3809 is dominated\nbelow $\\sim 2$ keV by a soft excess and from 2 to 10 keV by a hard\n($\\Gamma \\sim 1.3$) power law (Otani \\markcite{26} 1995).  In\ncontrast, a very steep spectrum with $\\Gamma_{(2-10keV)} \\sim 2.6$ was\nfound from NLS1 object RE~1034+39 (Pounds, Done \\& Osborne\n\\markcite{39} 1995).\n\nWe report the results from December 1993 {\\it ROSAT} and {\\it ASCA}\nobservations of Mrk~766. Timing analyses of two {\\it ROSAT} archival\nobservations are also presented. In section 2 the data reduction is\ndiscussed briefly.  In section 3 timing analyses using normalized\nvariability amplitudes and hardness ratios are presented.  In section\n4 the spectral analysis of the {\\it ASCA} data is described.  The\nresults are discussed in terms of standard models in Section 5 and\ncompared with reported results from other NLS1s.  A summary and\nconclusions are given in Section 6.\n\n\\section{{\\it ASCA} and {\\it ROSAT} Observations of Mrk~766}\n\nWe observed Mrk~766 with {\\it ASCA} and the {\\it ROSAT} PSPC during\nDecember 1993.  It had been previously observed with the {\\it ROSAT}\nPSPC several times. Two longer observations were made 1991 June 15 and\n1992 December 21 and these data were extracted from the {\\it ROSAT}\narchive. The observation log is given in Table 1.\n\nThe data were reduced using Xselect. To ensure all soft photons were\ncollected, {\\it ROSAT} extraction regions of $3 ^\\prime$ for the\non-axis 1993 observation and $4^\\prime$ for the off-axis 1991 and 1992\nobservations were used. Extraction of background subtracted light\ncurves from the events files was done using IDL software.  The light\ncurves from the off-axis 1991 and 1992 observations were corrected for\nvignetting. In the 1991 observation, Mrk~766 was periodically occulted\nby the detector rib so time periods in which the flux dropped to zero\nbecause of occultation were excluded (e.g. Brandt et al.\n\\markcite{2} 1993).  A region of the same radius located diametrically\nacross the detector and subjected to the same good time interval\nselection provided an approximately correctly normalized background.\n\n{\\it ASCA} data were reduced using standard selection and cleaning\ncriteria.  Background for the GIS was obtained from a source free\nregion in the GIS field of view approximately the same distance from\nthe optical axis as the source.  Background SIS spectra were obtained\nfrom blank sky fields, while background count rates for the light\ncurves were determined from the edges of the SIS chips. The time\ndependent gain shift in the SIS was accounted for by filling and using\nPI columns.  Spectra were extracted using Xselect, and background\nsubtracted light curves were obtained from the cleaned events files\nusing IDL.\n\nFigure~\\ref{fig1}a shows the {\\it ROSAT} PSPC and {\\it ASCA} SIS0\nlight curves from the most recent, quasi-simultaneous 1993\nobservations.  Error bars represent $1\\sigma$ statistical error. The\n{\\it ROSAT} observation started $\\sim 20,000$ seconds before the {\\it\nASCA} observation but was unfortunately cut short as the satellite\nwent into safe hold mode.  The {\\it ROSAT} PSPC light curves from the\n1991 and 1992 observations are shown in Figure~\\ref{fig1}b.  Mrk~766\nwas brightest in the {\\it ROSAT} band in 1992 at $<5 \\rm\\,\ncnts\\,s^{-1}$.  This is comparable to the flux level observed during\nthe {\\it ROSAT} All Sky Survey observation (Molendi, Maccacaro \\&\nSchaeidt \\markcite{20} 1993).  In order to make a direct comparison\nwith the other data, only the first $\\sim 80$ks of the 1992\nobservation was considered.  The remainder consists of data from only\ntwo orbits, is separated from the main part of the observation by\nnearly one day and the full light curve is shown in Netzer, George \\&\nTurner \\markcite{24} 1994.\n\n\\section{Timing  Analysis}\n\n\\subsection{Fastest Observed Variability}\n\nThe {\\it ASCA} light curves were examined in order to find instances\nof rapid variability that could be clearly identified in all four\ndetectors.  A dip in flux occurred at $\\sim 50,000\\rm \\, s$ from the\nstart of the observation; the light curve from the SIS0 detector is\nshown embedded in Figure~\\ref{fig1}a.  At the end of the dip, the flux\nincreased by nearly a factor of two in $\\sim 1000$ s. This timescale\nis the same order as those observed in {\\it ASCA} data from other\nSeyfert 1 nuclei, including NGC 4051 ($\\Delta t \\sim 200 \\rm \\,s$;\nMihara et al. \\markcite{18} 1994).\n\nAssuming that the X-ray emission originates from a single region, the\nminimum flux doubling time scale $\\Delta t$ gives an upper limit on\nthe source size as $R \\sim c \\Delta t$ of $3 \\times 10^{13}\\, \\rm cm$,\nwhere $c$ is the speed of light.  This estimate breaks down if the\nX-rays are emitted from many small regions.\n\nBecause of the telescope wobble, variability on timescales less than\n$400\\rm \\, s$ generally cannot be detected in {\\it ROSAT} data.\nSignificant variability was observed consistently between adjacent\norbits ($\\Delta t \\sim 6000$ s).  The fastest variability of the three\nobservations was found when it was brightest during the 1992\nobservation.  A 30\\% increase in 2400 s occurred 3400 s from the\nbeginning of the observation (Figure~\\ref{fig1}b).\n\n\\subsection{Variance analysis}\n\nThe normalized variability amplitude (NVA), defined to be the standard\ndeviation divided by the mean intensity, provides a simple way to\nquantify the variability in different energy bands (e.g. Edelson\n\\markcite{5} 1992).  If the width of the energy band is chosen so that the\nnumber of photons in each light curve is the same, the NVA collapses\nto the square root of the variance.\n\nThe {\\it ASCA} light curves from each detector were accumulated with\n100 s binning in the 4--10 keV band where the power law component\nshould dominate the spectrum.  The light curves were not background\nsubtracted.  However, the dilution of the variability by the constant\nbackground was small since the background rate in the highest energy\nband where contribution is largest was only e.g. $\\sim 8$\\% of the\nSIS0 count rate.  The data below 4 keV were divided into energy bands\nwith bounds chosen so that the light curves had the same mean square\nmeasurement error $\\sigma^2_{err}$ as in the 4--10 keV band.  Note\nthat each resulting band was wider than the energy resolution at that\nband.  The true variance of the data given by\n$\\sigma^2_{int}=\\sigma^2_{obs}-\\sigma^2_{err}$ is plotted as a\nfunction of energy in Figure~\\ref{fig2}.  The $1\n\\sigma$ uncertainties in the variance are less than 10\\% of the values\n(Equation 2 of Done et al.  \\markcite{4} 1992).  These variance plots\ndemonstrate that the variability amplitude over the whole observation\nis smallest at hard energies, peaks near 1 keV, and decreases towards\nlower energies in the SISs.\n\nFor the {\\it ROSAT} light curves, three standard energy bands were\nconsidered: the `{\\it a} band' (channels 11--41, energy 0.1--0.4 keV),\nthe `{\\it c} band' (channels 52--90, energy 0.5--0.9 keV) and the\n`{\\it d} band' (channels 91--201, energy 0.9--2 keV). These bands are\nindependent and roughly correspond to regions where different spectral\nfeatures will dominate: the soft excess in the {\\it a} band, the warm\nabsorber in the {\\it c} band, and the power law in the {\\it d} band.\nThe binning of the background subtracted light curves was chosen to be\n400~s to account for the telescope wobble.  \n\nBecause of the rib occultations in the 1991 data, it was necessary to\ninclude data with net exposure per bin of 200--400~s, even though the\nshorter exposure bins add noise to the light curve. The light curves\nin these three bands are shown on the left in Figures~\\ref{fig3}a, b\nand c.  Mrk~766 is bright enough and the bin size is long enough that\nthe signal to noise is better than 6 in all bins. Variability was\ndetected in each energy band and the $\\chi^2$  values for a constant\nhypothesis model fit and the computed NVAs are listed in Table 2. The\nNVAs show that in all observations the source is significantly less\nvariable in the lowest energy band compared with the higher energy\nbands.\n\n\\subsection{Flux Ratios}\n\nThe hardness ratio (4.0--10.5 keV\/1.0--1.35 keV) and softness ratio\n(0.4--0.7 keV\/1.0--1.35 keV) light curves from the {\\it ASCA} SIS0\ndetector are shown in Figures~\\ref{fig4}a and 4b, respectively.\nFollowing Ptak et al.  \\markcite{27} (1994) the values were computed\nusing variable bin sizes corresponding to good time intervals longer\nthan 300 seconds.  A large decrease in hardness corresponding to a\nsoftening of the spectrum was observed in the hardness ratio light\ncurve at about 20,000~s from the beginning of the observation\n(Figure~\\ref{fig4}a).  The light curves show that the spectral change\nis due to a large increase in 1.0--1.35 keV flux while the 4--10 keV\nflux remained nearly constant.  Significant variability in the\nsoftness ratio was also observed at the same time, such that the\nspectrum below 1 keV hardens when the flux increases\n(Figure~\\ref{fig4}b).  The hardening of the spectrum may be more\ngradual and it is not clearly completed until after $\\sim 25,000$~s.\nVariability was observed in both energy bands but the amplitude is\nlarger in the 1.0--1.35 keV band.  The spectral variability is\nconfined to the region around 20,000~s elapsed time.  Other instances\nof large amplitude flux variability occurred (e.g. at $\\sim 35,000$ s)\nwith no corresponding spectral change.  Thus, the spectral variability\nis not strictly correlated with the flux.\n\n{\\it ROSAT} softness ratio light curves (0.1--0.4 keV\/0.9--2.0 keV and\n0.5--0.9 keV\/0.9--2.0 keV) are shown on the right sides of\nFigures~\\ref{fig3}a, b and c.  Spectral variability is most pronounced\nbetween the hardest and softest bands, with the spectrum generally\nhardening as the flux increased.  This result is also true from\nobservation to observation; the spectrum was hardest when the source\nwas brightest in 1992, and softest when it was dimmest in 1991.  On\nlong time scales the softest band varies proportionately less than the\nharder bands.  On short time scales the spectral variability is not\ncorrelated with the flux.  At high flux, when the source was most\nrapidly variable, correlated variability occurred with no change in the\nhardness ratio (1992: at the beginning of the observation).  Similar\ncorrelated variability was observed during the {\\it ROSAT} All Sky\nSurvey when the source was also quite bright (Molendi, Maccacaro and\nSchaeidt \\markcite{20} 1993).  In contrast, at low flux in the 1991\nand 1992 observations, orbit to orbit deviations in the hardest band\noccurred which were not followed in the softest band (1991: at $\\sim\n23000$ and $\\sim 58000$~s; 1992: at $\\sim 66000$~s).  These short time\nscale hard band excursions result in dips in the softness ratio.\n\n\\section{Spectral Analysis}\n\nBased on the softness and hardness ratios, the {\\it ASCA} data were\ndivided in two as shown in Figures~\\ref{fig4}a and 4b, and spectra were\naccumulated avoiding the transition region between $\\sim 15,000$ and\n$\\sim 25,0000$~s from the beginning of the observation.  These spectra\nare referred to as the low state and high state spectra and represent\nexposures of $\\sim 8,000$ and $20,000$~s respectively.\nFigure~\\ref{fig5} shows the pha ratios from the low state divided by\nthe high state of the summed SIS0 and SIS1 spectra in the top panel\nand of the summed GIS2 and GIS3 spectra in the bottom panel.  These\nconfirm the results of the variance analysis which were that the\nspectrum is most variable at around 1 keV, the variability amplitude\ndecreases to lower and higher energies, and the flux is essentially\nconstant at about 10 keV.\n\n\\subsection{The Hard X-ray Spectrum}\n\nAn estimation of the power law index, assuming that any warm absorber\ndoes not have a large column density, was obtained by fitting the four\nspectra above 2 keV with a power law plus iron $K\\alpha$ line model.\nThe low and high state spectral indices were $1.57\\pm 0.07$ and\n$2.00^{+0.03}_{-0.04}$ respectively, and the results from fitting each\ndetector separately were consistent.  Throughout this paper, quoted\nuncertainties are determined assuming 90\\% confidence and 1 parameter\nof interest ($\\Delta \\chi^2=2.71$).\n\n\\subsection{The Soft X-ray Spectrum}\n\nThe shape of the soft X-ray spectrum can be seen by fixing the power\nlaw indices to the values found above 2 keV, including the Galactic\nabsorption ($1.77 \\times 10^{20} \\rm \\, cm^{-2}$; Elvis, Wilkes \\&\nLockman\n\\markcite{6} 1989), and plotting the ratio of the data to the model.\nThese plots are shown in Figure~\\ref{fig6}a and 6b for the SIS0 and\nGIS2 low state and high state spectra respectively.  In the low state\nthere is excess emission below 0.8 keV, suggesting the presence of a\nsoft excess component (e.g.  Mihara et al. \\markcite{18} 1994).  In\nthe high state there is a deficit between 0.8 and 1.3 keV, suggesting\nthe presence of a partially ionized absorber (e.g. Fabian et al.\n\\markcite{8} 1994).  It is clear that an absorbed power law plus\nnarrow iron line cannot model the spectra.  Spectral fitting results\nof the SIS0:GIS2 and SIS1:GIS3 pairs are listed in Table 3 for low and\nhigh states separately.  The iron line is discussed separately in\nSection 4.5 and fit results are given in Table 4.\n\nThe soft excess component and warm absorber can be simply\nparameterized using a black body model and an absorption edge,\nrespectively.  The results from adding each of these components\nseparately are given in the second and third panels of Table 3.\nAddition of either model component significantly improved the fits of\nboth the low and high flux spectra.  However, the addition of the\nblack body improved the fit more than the edge in the low flux case,\nwhile the reverse was true for the high flux case, as expected from\nthe residuals shown in Figure~\\ref{fig6}.\n\nThe fourth panel of Table 3 lists results from fitting with a model\nincluding both a blackbody and an edge.  These fit results confirm the\nimportance of the black body component in the low state spectra and\nthe edge in the high state spectra. The black body temperatures and\nedge energies of the low and high state spectra are consistent. The\nedge energy near $0.74\\rm \\, keV$ is consistent with an origin of\ntransmission through a partially ionized absorber dominated by\n\\ion{O}{7}.  The power law slope of the low flux spectra remains\nsignificantly flatter than that of the high flux spectra ($\\Delta\n\\Gamma \\sim 0.35$).\n\nNext, to better simulate the warm absorber, a two edge plus power law\nmodel was tried and the results are listed in the fifth panel of Table\n3.  The power law indices are steep in both the low and high state,\nand the implied change in index is much smaller ($\\Delta\\chi^2\n\\sim 0.1$).  This model has the same number of degrees of freedom as\nthe power law plus black body and edge model, but the fit is much\npoorer for the low flux spectra ($\\Delta \\chi^2$ of 24 and 50 for the\nSIS0:GIS2 and SIS1:GIS3 pairs, respectively).  In contrast, the fits\nof the high flux spectra are improved somewhat by the additional edge\n($\\Delta \\chi^2$ of 9 and 11).  Addition of a black body component,\nwhile not necessary for the high flux spectra, greatly improves the\nfit of the low flux spectra ($\\Delta \\chi^2$ of 28 and 52), and\nresults in a decrease in the low flux photon index and an increase in\nthe implied index change ($\\Delta\\Gamma \\sim 0.35$).  The temperature\nof the black body component in the low and high states is consistent\nat $kT \\sim 120\\rm eV$, although the limits on the temperature in the\nhigh state cannot be determined well for the SIS1:GIS3 pair.  The edge\nenergies near 0.74 and $0.87\\rm \\, keV$ found in the high state are\napproximately consistent with absorption by \\ion{O}{7} and \\ion{O}{8}.\nIn the low state, the \\ion{O}{7} edge is clearly detected but the\nsecond edge is not necessary.  This suggests that the ionization of\nthe gas in the high state is higher than in the low state.\n\nWe also modeled the warm absorber using a table model (e.g.  Yaqoob,\nWarwick \\& Pounds \\markcite{37} 1989).  The model used here assumes\nthat the power law is the sole source of ionizing photons. The density\nof the gas was assumed to be $10^{9.5} \\rm cm^{-3}$.  An analytic\napproximation based on a large number of CLOUDY runs was used to model\nthe temperature as a function of ionization parameter.  It was found\nto range between $\\sim 5 \\times 10^4$K and $\\sim 10 \\times 10^4$K. The\nadvantages of using the warm absorber table model compared with the\ntwo edge model are that there are two fewer parameters, and the\nparameters ($log(U)$ and $log(N_w)$) can be directly interpreted.  The\ndisadvantage is that we must assume a particular model for the\nionizing spectrum and gas.  Further, in this model, only absorption is\nconsidered, and the emission lines expected if the warm absorber has a\nlarge covering factor are ignored (e.g. Netzer \\markcite{23} 1993).\nHowever, this model is sufficient for a general discussion given the\nstatistical quality of these data (but see Section 4.3.1).\n\nThe four groups of spectra were first fit with a power law plus warm\nabsorber model (Table 3).  As found using the two edge model, the fits\nof the high flux spectra are acceptable, but the fits of the low flux\nspectra are unacceptable.  Addition of a black body component\nsubstantially improves the fit of the low flux spectra, as shown in\nthe eighth panel of Table 3, but it is not necessary to model the high\nflux spectra.  The ionized column density is consistent between the\nlow and high flux states at $log(N_w) \\sim 21.8$, while the ionization\nstate $\\log(U)$ is lower for the low flux spectra ($-0.78$) compared\nto the high flux spectra ($-0.4$).  In this model, ionization states\n$log(U)$ below and above $\\sim -0.4$ are dominated by \\ion{O}{7} and\n\\ion{O}{8} absorption, respectively. In contrast with the two edge\ndescription of the warm absorber, the use of the warm absorber table\nmodel resulted in a higher black body temperature in the high flux\nstate; however, the black body was barely detected in the high flux\nspectra ($\\Delta \\chi^2$ of 4 and 5).  Thus the temperature of the\nsoft component is model dependent, but the necessity of this component\nto model the low flux spectra is not. The table model description of\nthe warm absorber results in a slightly steeper index compared with\nthe two edge description.  This is because the table model properly\ntreats the absorption by gas with cosmic abundances (i.e. not only\noxygen) producing curvature in the model between 1.5 and 2.5 keV.\nThus the value of the indices is slightly model dependent, but the\nchange in index between the low and high states is not.\n\nIn summary, these results show that to model the high and low flux\nspectra consistently, both a soft excess component and a warm absorber\nare required.  When both of these components are included in the\nspectrum, the low flux spectral index is significantly flatter than\nthe high flux index ($\\Delta\\Gamma \\sim 0.35$).\n\n\\subsection{Other Models}\n\n\\subsubsection{Other Soft Excess Models}\n\nThe soft excess component in the low flux spectra was modeled\nadequately with a black body (with a single edge, $\\chi^2 = 230$\/266\nd.o.f. for the SIS0:GIS2 pair).  For comparison, other soft excess\nmodels including Raymond--Smith (cosmic abundances), bremsstrahlung,\ndisk black body, and power law were tried.  The power law plus soft\ncomponent model alone did not fit the low flux spectra well.  The disk\nblack body model fit the best, with $\\chi^2$  of 247\/268 d.o.f.  Including\nan edge generally improved the fits, but the Raymond--Smith and the\npower law models could not describe the low flux spectra well ($\\chi^2$  of\n277 and 259\/266 d.o.f. respectively), while the bremsstrahlung and the\ndisk black body models could ($\\chi^2$  of 234 and 231\/266 d.o.f.\nrespectively).  A slightly higher temperature was found using these\nmodels ($kT=200$ eV and $kT=145$ eV) compared with the black body\nmodel ($kT=117$ eV) but nearly consistent edge energy, edge depth and\nphoton index were found.  The indices obtained were flat ($\\Gamma =\n1.57$ in both cases).\n\nThe warm absorber table model used thus far models only the absorption\nedges of various ionized species.  Since the line emission expected\nfrom a physical warm absorber might appear as an excess emission\ncomponent, we also tested models calculated using XSTAR which include\nemission lines from the warm absorber (Kallman \\& Krolik\n\\markcite{16} 1993).  Fitting the SIS0:GIS2 pair with a power law\nplus emission only from a physical warm absorber in the line of sight\nresulted in a better fit with $\\chi^2$  of 292\/277 d.o.f.  Including an\nedge, to simulate the absorption by a warm absorber, resulted in a $\\chi^2$ \nof 261\/268 d.o.f.  However, a significantly better fit was found when\na blackbody was also included ($\\chi^2$  = 228\/264 d.o.f.) and the resulting\npower law index was again flat ($\\Gamma = 1.55$).  Similarly, emission\nfrom reflection by a warm absorber could not alone describe the\nspectrum (alone: $\\chi^2=299\/268$ d.o.f.; with an edge:\n$\\chi^2=268\/266$ d.o.f.; with an edge and a black body:\n$\\chi^2=227\/264$ d.o.f.).  In the final case, the index was flat\n($\\Gamma = 1.55$).\n\nSoft excess emission can also be produced by reflection from an\nionized disk (Ross \\& Fabian \\markcite{28} 1993; Zycki et al.\n\\markcite{38} 1994).  We fit the low flux spectrum with an ionized\ndisk table model computed according to Zycki et al. \\markcite{38}\n1994.  For power law plus disk emission only the $\\chi^2$  was 262\/269\nd.o.f., but the ratio of the reflected flux to direct emission was\n5.8.  Since the ratio should be near 1 for the isotropic static case.\nwe considered this result to be unphysical.  Addition of an edge gave\n$\\chi^2$  of 248\/267 d.o.f., but again the ratio was too large at 5.5.\nAddition of a black body gave $\\chi^2=228\/265$ d.o.f., with the ratio\nreduced to a physical value of $R=1.23$ and a flat index $\\Gamma =\n1.61$.\n\nThese results show that soft excess models without line emission fit\nthe spectra well, but we cannot distinguish among them, possibly\nbecause of the poor statistics due to the relatively short exposure in\nthe low state and the decrease in sensitivity toward low energies of\nthe {\\it ASCA} SIS.  Further, the flat spectral index obtained in the\nlow state is robust against changes in the soft excess model.\n\n\\subsubsection{Partial Covering Models}\n\nAnother possible origin of soft emission is leakage through a\npartially covering absorber (e.g. NGC 4151: Weaver et al.\n\\markcite{35} 1994).  The spectral variability would then result from\na change in the covering fraction.  However, the partial covering\nmodel does not fit the low flux SIS0 and GIS2 spectra well\n($\\chi^2=298$ for 268 d.o.f.), the resulting power law is forced to be\nsteep ($\\Gamma=2.42$), and both low and high energy residuals are\nseen.  These can be modeled by reflection, in which case the fit is\ngood ($\\chi^2=225$\/263 d.o.f.), but the power law is very steep\n($\\Gamma=3.0$) and the ratio of the reflected emission to primary\nemission is required to be 20.  This model is unphysical so we\nconclude that partial covering cannot adequately model the soft excess\ncomponent. Finally, a decrease in the fraction of the source covered can\nonly produce a steepening of the spectrum with an increase in flux and\ncannot explain the hardening of the spectrum below 1 keV indicated by\nthe softness ratio (Figure~\\ref{fig4}b).\n\nA scattering and dual absorber model was used to describe the complex\nX-ray spectrum of NGC 4151 (Weaver et al.  \\markcite{35} 1994).  For\nMrk~766 this model does not give a good fit ($\\chi^2=244$\/266 d.o.f.)\nand the photon index is steep ($\\Gamma=2.84$). Low energy residuals\nsuggest an unmodeled absorption edge.  When an edge is included the\nfit is good ($\\chi^2$ =222\/264 d.o.f.), but the power law index is very\nsteep ($\\Gamma=2.94$).  Again, this steep index seems unphysical, so\nthe dual absorber model is rejected.\n\n\\section{Reflection}\n\nThe spectral index change is robust against changes in models of the\nsoft excess component and warm absorber because the low flux spectrum\nis flat at high energies.  The reflection spectrum is also flat (e.g.\nGeorge \\& Fabian  \\markcite{9} 1991), and could produce a hard tail if\nthe reflection component normalization is high compared with the power\nlaw normalization.  This could be observed if the response of the\nreflection component lags variability of the incident X-rays and would\nbe expected if the light crossing timescale of the reflection region\nis long compared with the source variability timescale, or if the\nreflection region is located far from the X-ray source (e.g. in the\nmolecular torus; Ghisellini, Haardt\n\\& Matt \\markcite{12} 1994; Krolik, Madau \\& Zycki\n\\markcite{17} 1994; Leighly et al. \\markcite{42} 1996).\n\nWe define the reflection ratio to be 1 under the conditions that\nnonvarying primary power law emission from an isotropic point source\nilluminates an infinite optically thick disk.  In this case, the\nreflection spectral component is not important in the spectrum below 5\nkeV.  Thus, we can estimate the contribution of the reflection by\nfitting the spectra below 5 keV and comparing with the fit results\nover the full range.  A power law plus two edge fit of the SIS0:GIS2\nspectra results in a steep photon index (low state: $\\Gamma=1.95$ and\n$\\chi^2$  = 214.4\/230 d.o.f., high state:  $\\Gamma=2.02$ and $\\chi^2$ =524\/519\nd.o.f.).  Addition of a black body component to the model improved the\nfit of the low flux spectra and flattened the photon index\n($\\Gamma=1.67$ and $\\chi^2=197\/228$ d.o.f.), but had no effect on the\nhigh flux spectral fit.  If the table model is used to model the warm\nabsorber, similar results are obtained although both indices are found\nto be slightly steeper.  Thus, fitting below 5 keV shows that the\nindex variability is still required and also that the measured low\nenergy index is consistent with the data above $\\sim 5 \\rm keV$.\n\nA reflection ratio much larger than 1 results in a flat spectrum\ntowards high energies.  Fitting the spectra with a power law plus 2\nedges and reflection allowing the ratio to be free produces a good fit\nwith a steep index which is consistent between the low and high flux\nspectra (Table 3).  However, to explain the low flux spectrum, the\nreflection ratio must be 5--8, while the high flux spectra require a\nreflection ratio of only 0.5.  Most importantly, the reflection\ncomponent normalization is required to be significantly {\\it higher}\nin the low flux state compared with the high flux state, so a\nreflection lag cannot explain the spectral variability.  Further, such\na large reflection ratio should be accompanied by a large equivalent\nwidth narrow iron line in the low state spectrum and such a line is\nnot observed (Section 4.5).\n\nThe timing analysis results also rule out a lag in neutral reflection as\nthe origin of the spectral variability.  Since the neutral reflection\nspectrum flux decreases towards low energies, only spectral softening\nwith an increase in power law flux is predicted, whereas a hardening\nof the spectrum below 1 keV was observed. If the surface of the\nreflector is ionized, the opacity is reduced at low energies and a\nsoft X-ray reflection component plus emission lines should be observed\n(Ross \\& Fabian \\markcite{28} 1993; Zycki et al. \\markcite{38} 1994),\nwhich results in hardening of the low flux spectrum with an increase\nin power law flux.  It was shown in Section 4.3.1 that the ionized\ndisk model does not produce a good fit alone, mostly because the soft\nexcess does not show evidence for line emission.\n\n\\subsection{The Iron Line}\n\nThe presence of the iron $K\\alpha$ line in Seyfert 1 nuclei spectra\nis well established (e.g. Nandra \\& Pounds \\markcite{22} 1994).  Mrk\n766 was not observed using Ginga, and a iron $K\\alpha$ line had not\nyet been observed in its spectrum.  To look for the iron line, we fit\nthe spectra from all four detectors simultaneously above 2 keV.\nBecause of the continuum spectral variability the low and high state\nspectra were fit separately.  A power law model resulted in\n$\\chi^2$\/d.o.f. of 206\/253 and 806\/759 for the low and high states\nrespectively.  Addition of a narrow ($\\sigma=0$ keV) redshifted line\nwith the energy fixed at 6.4 keV gave the results presented in Table\n4.  There is no strong evidence for a narrow line with rest energy 6.4\nkeV in the low flux spectra ($\\Delta\\chi^2 \\sim 2$). In the high flux\nspectra $\\Delta\\chi^2$ is 13 corresponding to an F statistic value of\n12.4, indicating the presence of a line with confidence greater than\n99.9\\%. Thus the presence of an iron emission line is confirmed in\nthis source.  The ratio of data to power law model for the summed SIS0\nand SIS1 spectra shows the narrow line and ionized iron edge\n(Figure~\\ref{fig7}).  The non-detection in the low flux spectra can be\nexplained by the poor statistics resulting from lower flux and shorter\nexposure.  If the high flux data are divided into several spectra\ncharacterized by shorter exposures, the presence of a line in the\nspectra from all detectors separately cannot be confirmed.  Next, the\nline energy was allowed to be free.  The high state line energy is\nconsistent with 6.4 keV and a lower energy line is marginally detected\nin the low state spectra.  The line energy in both states is\nconsistent with an origin in primarily neutral material, and emission\nfrom highly ionized material is excluded. The line equivalent width is\n$\\sim 100 eV$ consistent with that expected from emission from an\naccretion disk (e.g. George\n\\& Fabian  \\markcite{9} 1991).  Broad iron lines have been discovered in the\n{\\it ASCA} spectra of several AGN (e.g.  Mushotzky et al.\n\\markcite{21} 1995).  A broad line is preferred in the low flux\nspectra, although the addition of another parameter reduces the $\\chi^2$  by\nonly 5, so again the detection is marginal. The line is narrow in the\nhigh flux spectra, constrained with $\\sigma < 0.2\\,\\rm keV$.  The\nshape of the high energy continuum changes the measured line\nparameters.  When reflection with ratio fixed to 1 is included,\nsimilar results were obtained as before, but the measured equivalent\nwidths were smaller (Table~4).\n\nThese results also support the hypothesis that a lag in the reflection\ncomponent cannot be the origin of the spectral variability.  A lag\nimplies that the bulk of the reflected emission should come from a\nsubstantial distance from the source, and so the line would be\nexpected to be narrow.  However, the large reflection ratio required\nto fit the low flux spectra predicts a very large equivalent width ($>\n\\sim 500\\rm eV$) narrow line, which would be easily detected if\npresent.\n\n\\subsection{Quantifying the Spectral Variability}\n\nThe results of the previous sections indicate that the power law with\nblack body and warm absorber model provides the best fit to both the\nlow and high flux spectra.  A change in spectral index seems to be\nindicated.  However, since the model is complex, spectra from the two\nstates must be fit simultaneously to determine which parameters\nnecessarily change.  Neutral absorption, originating in our Galaxy and\nthe host galaxy, was assumed not to change, and the iron line was\nmodeled as narrow with fixed energy.  Combined fits were done using both\ndescriptions of the warm absorber.\n\nWhen two edges were used to model the warm absorber, the edge energies\nwere equated in the high and low state models.  This was done because\nthe low state spectra could not constrain the higher edge energy and\nsince the edges are identified as \\ion{O}{7} and\n\\ion{O}{8} edges, no change is expected in the energies.  This model fit the\nlow and high spectra well ($\\chi^2$  of 860.1\/859 d.o.f. and 789.8\/844\nd.o.f. for the SIS0:GIS2 and SIS1:GIS3 spectra respectively).  The\ntemperatures of the black body were consistent between the low and\nhigh states so these were equated resulting in a negligible increase\nin $\\chi^2$.  No other parameters could be equated without resulting\nin a large change in $\\chi^2$.  The results are listed in the top\npanel of Table~5, and they indicate that a significant change in index\nand black body normalization occurred.  The edge energies are roughly\nconsistent with absorption by \\ion{O}{7} and \\ion{O}{8}.  The optical\ndepths of these edges are consistent between the low and high state,\nso no change in the ionization of the warm absorber can be determined\nfrom these fits.  However, the best fit value of $\\tau_{OVII}$ is\nlarger in the low state than in the high state, and the reverse is\napproximately true for $\\tau_{OVIII}$.  This suggests that an increase\nin the ionization occurred, but the statistics are too poor to require\nthis conclusion.\n\nThe results of the combined fits using the warm absorber table model\nare listed in the second panel of Table 5.  The ionized column\ndensities were consistent so they were equated in the spectral\nfitting.  As noted previously, when the warm absorber was described\nusing the table model, the black body temperature was found to be\nhigher in the high state than in the low state (Table 3).  When the\ntemperatures are equated in the combined fits, the increase in\n$\\chi^2$ is 2.7 and 5.7 for the SIS0:GIS2 and SIS1:GIS3 respectively,\nsignificant with 90\\% and 97.5\\% confidence. However, this effect is\nclearly model-dependent and may be due to the shape of the warm\nabsorber model and possibly the absence of emission lines in the\nmodel, and thus the implied black body temperature change is unlikely\nto be physical.  Three parameters changed between the low and high\nstates: the power law index, the black body normalization, and the\nionization parameter.  As these parameters are coupled in the spectral\nfitting, to evaluate the significance of the change, the $\\chi^2$  contours\nwere plotted for each pair of parameters (Figure~\\ref{fig8}a, b, and\nc).  These show that the results are consistent between the SIS0:GIS2\nand SIS1:GIS3 spectral pairs, and that the index change, the\nionization state change and the black body normalization change are\nsignificant with $>99$\\%, 90\\% and 68\\% confidence respectively.  We\nnote that the change in the ionization state is a model dependent\nresult, as we cannot demonstrate a change in the optical depths of the\ntwo oxygen edges.  Figure~\\ref{fig9}a and b show the best fitting\nmodels, spectra, and ratios between spectra and model for the low and\nhigh state SIS0:GIS2 spectra.  Figure~\\ref{fig10} shows the best fit\nmodels for the low and high state SIS0 spectra.  Note that the pivot\npoint for the power law change is $\\sim 9$ keV.\n\n\\subsection{{\\it ROSAT} Spectral Fitting}\n\nThe {\\it ASCA} data show that the soft X-ray spectrum is complex,\ncomprised of a power law with variable index, a warm absorber with\nvariable ionization state and a black body with marginally variable\nnormalization and model dependent temperature.  Seven parameters are\nrequired to describe the spectrum in the {\\it ROSAT} band.  Because of\nthe poor spectral resolution, the PSPC spectra have 5 independent\nchannels.  Thus, detailed fitting of the PSPC spectra is of limited\nvalue, as multiple models are acceptable.  Qualitatively, the spectra\nfrom the 1992 and 1993 observations are very soft and cannot be\nadequately modeled using a single power law plus absorption.  A soft\ncomponent like a black body gives a good fit, with $kT\n\\sim$70--90 eV, and a power law with index $\\sim $1.9--2.0.  An edge\ncan also model the spectrum, but the spectral index is steeper at\n$\\sim 2.5$.\n\n\\section{Discussion}\n\n\\subsection{The Hard Spectral Variability}\n\nThe most significant result of this study was the observation of\ndramatic photon index variability from $\\sim 1.6$ to $\\sim 2.0$, over\nseveral thousand seconds and confined to a single event.  We discuss this\nresult in light of current models of the X-ray power law emission in AGN.\n\n\\subsubsection{General Considerations}\n\nThe high energy power law observed from AGN can be successfully and\nplausibly explained by inverse Comptonization by high energy electrons\nof soft UV photons likely originating in the accretion disk. The rapid\nX-ray variability of some AGN implies a high radiation density in the\nnucleus which results in production of electron-positron pairs. The\nimportance of pair production is determined by the compactness\nparameter, $$l= L\\sigma_T\/R m_e c^3,\\eqno(1)$$ where $L$ is the\nluminosity, $R$ is the source size, $\\sigma_T$ is the Thompson\nscattering cross section, $m_e$ is the mass of the electron, and $c$\nis the speed of light.  If the compactness is high, the optical depth\nto pair production will exceed unity and pairs will be produced which\ncan substantially modify the emerging spectrum.  Generally speaking,\nthese models can be differentiated by whether the high energy\nelectrons are thermal or accelerated by non-thermal processes, since\npair production limits the highest energy attainable in the thermal\nplasma.  In rapidly variable AGN, however, both thermal and\nnon-thermal processes may be present (Ghisellini, Haardt \\& Fabian\n\\markcite{10} 1993).\n\nThe results presented here suggest that Mrk~766 is compact enough that\npairs should be produced in the nucleus.  The X-ray flux was observed\nto change by a factor of two in $\\sim 1000$ seconds. This corresponds\nto a source size upper limit of $R < c\\Delta t \\sim 3 \\times 10^{13}\n\\rm \\, cm$.  The 2--10 keV luminosity is $1.3 \\times 10^{43}\\, \\rm ergs \\,\ns^{-1}$ in the high state, implying an X-ray compactness parameter of\n$l_x \\sim 12$.  The hard compactness parameter, proportional to the\ntotal luminosity in the hard component, could be substantially larger.\nIf the compactness parameters are larger than 10, pair production\nshould be important if there are an adequate number of $\\gamma$-ray\nphotons present.  In non-thermal models, the $\\gamma$-rays are\nproduced through upscattering of soft photons by extremely\nrelativistic electrons.  In thermal models, the origin is primarily the\nhigh energy tail of the thermal spectrum and thus the number of\n$\\gamma$-rays depends on the temperature of the plasma. OSSE\nobservations of a few AGN find that the temperature is large enough\nthat electron positron pairs should be produced (e.g. see Figure 1 of\nFabian  \\markcite{7} 1994); however, there have been no high energy\nspectra obtained from Mrk~766.\n\n\\subsubsection{Simple Thermal Comptonization Models}\n\nIf the source does not contain many pairs, and if the power law\nresults from unsaturated Comptonization of soft photons, the spectral\nparameters are very simply related to one another (e.g. Rybicki \\&\nLightman \\markcite{29} 1979).  The slope of the power law depends\ninversely on the Compton $y$ parameter which is proportional to a\npower of the temperature, so an increase in temperature implys a\ndecrease in index.  However, if the soft photon input is constant, the\npivot point energy of the photon index change should be the energy of\nthe soft input photons.  In contrast, we observe the pivot point to be\nmuch higher, at $\\sim 9$ keV.\n\nIf the thermal plasma is pair dominated, Ghisellini \\& Haardt (1994)\n\\markcite{11} show that there is a one--to--one mapping of the\nobservables ($kT$ and $\\alpha$, where $\\alpha=\\Gamma-1$ is the energy\nindex) to the plasma parameters ($l_H$ and $l_H\/l_S$, where $l_H$ and\n$l_S$ are the hard and soft compactnesses, characteristic of the\nrelativistic electrons and the soft (UV) seed photons, respectively).\nOur observed increase in photon index by $\\Delta \\alpha \\sim 0.4$\nimplies a decrease in $l_H\/l_S$ by a factor of 10 (Figure 2 of\nGhisellini \\& Haardt \\markcite{11} 1994).  We observed a 2--10 keV\nflux increase by a factor of 1.3, but since the power law pivot point\nis $\\sim 9\\rm keV$, integration to high energies may show that the low\nflux state luminosity is actually larger than the high flux state\nluminosity.  OSSE observations of several Seyfert 1 galaxies have\nfound that the power spectrum is cut off above several hundred keV\n(e.g. Fabian \\markcite{7} 1994).  Integration of the power law from\n2~keV to the generous upper limit of 500~keV shows that the low state\n$l_H$ is at most a factor of 2 larger than the high state $l_H$,\npredicting an increase in index by only $\\sim 0.1$.  Further, the time\nscale of the spectral variability, less than 10,000 seconds, precludes\na large increase in $l_S$, since this short time scale would be the\norder of the orbital period at the innermost stable orbit for a $< 5\n\\times 10^{7} M_\\odot$ black hole.  A change in the accretion rate\nshould be characterized by the viscous or radial drift time scale,\nestimated by Molendi \\& Maccacaro (1994) \\markcite{19} to be 2.6 days.\nFurther, the {\\it ROSAT} spectral variability can be most naturally\nexplained by a constant (on time scales of one day) soft component\ndominating the softest X-ray band (see Section 5.2).  Finally, as\nGhisellini \\& Haardt (1994) \\markcite{11} note, if reprocessing in the\ndisk is important, $l_H\/l_S$ would be expected to remain approximately\nconstant, and little intrinsic index variability should be observed.\nHowever, this model is very simple, and predictions may change\nsubstantially if a realistic geometry or self-consistent treatment of\nthe two phases is considered.\n\n\\subsubsection{Nonthermal Comptonization Models}\n\nStatic nonthermal models of X-ray emission have been investigated by\nseveral authors (e.g. Svensson \\markcite{31} 1994 and references\ntherein) and 2--10 keV spectral index variability has been studied by\nYaqoob \\markcite{36} (1992). Most simply and generally, soft photons\nwith dimensionless frequency $x_S$ and compactness $l_S$ are scattered\nby relativistic electrons with Lorentz factor $\\gamma_0$ and\ncompactness $l_H$.  First order scattering produces a flat photon\nspectrum with $\\Gamma \\sim 1.5$ extending to\n$x_{max,1}=max[4\/3\\gamma_0^2 x_s,\\gamma_0]$ (Svensson \\markcite{30}\n1987). Pairs are produced if the photon spectrum extends to\nsufficiently high energies and if the optical depth to pair production\nis greater than unity.  Soft photons reprocessed by pairs have a\nsteeper spectrum with $\\Gamma=1.75$ breaking sharply at $x_B=2\n\\gamma_0^4 (2\/3 x_s)^3$. If the energy of pair reprocessed photons is\nrelatively low, they could be observed as an X-ray soft excess.  If\nthe energy of reprocessed photons is high enough, additional pair\ngenerations will be produced resulting in a pair cascade.  In that\ncase, the photon spectrum is steep with $\\Gamma$ approaching 2 (e.g.\nSvensson \\markcite{30} 1987).\n\nA nonthermal model can naturally explain the observed change in index,\nthe disappearance of the soft excess component and the confinement of\nthe spectral variability to a single event. The photon index\nvariability could result from a transition from a first order pair\nspectrum to a cascade caused by a sudden increase in the Lorentz\nfactor of the relativistic electrons.  Thus, the low flux spectrum is\ncomprised of the inverse Compton cooling spectrum, characterized by\nthe hard X-ray power law with photon index near $1.5$, and the first\norder optically thin pair reprocessed spectrum, observed as the soft\nexcess component.  The high flux spectrum is comprised of a pair\ncascade spectrum, characterized by the hard X-ray power law with\nphoton index near 2.  The soft excess component disappears in the high\nflux spectrum as the maximum energy of the pair reprocessed spectrum\nincreases far beyond the observed X-ray band, resulting in the\ncascade.  The change in flux in the X-ray band depends on the change\nof hard compactness $l_H$ which can be expected to accompany the\nchange in Lorentz factor, and flux variability uncorrelated with\nspectral variability would occur through variation in $l_H$ alone.\n\nA possible difficulty with non-thermal models which produce flat\nspectra is that, in general, they overpredict the gamma ray\nbackground.  However, recent work shows that if the non-thermal plasma\nis in a corona above a disk, $\\gamma$-ray photons are more efficiently\ndepleted and, depending on the compactness, a strong spectral cut off\nbelow 200 keV is predicted (Tritz \\markcite{32} 1990; Tsuruta \\&\nKellen \\markcite{40} 1995).  \n\n\\subsection{The Soft Spectral Variability}\n\nThe {\\it ROSAT} spectrum was observed to become harder as the flux\nincreased, consistent with the lower amplitude variability observed at\nsoftest energies. A most natural explanation for this behavior is\nvariability between the relative normalizations of the power law and\nsoft excess component.  This would be observed if the flux of the soft\ncomponent were nearly constant on the time scale of an observation.\nIn terms of current physical models it could imply that the soft\ncomponent is dominated by primary emission from an accretion disk and\nreprocessed hard emission is relatively less important.  The fraction\nof black body flux in the softest band can be estimated by comparing\nthe relative variability of the hardest band where the power law\ndominates with the relative variability of the softest band where the\nsoft component dominates.  This scenario predicts that if a static\nsoft excess component comprises a large fraction of the flux in the\nsoftest band, any flux change must be accompanied by a hardness ratio\nchange, while if the black body comprises only a small fraction of the\nflux, variability in hard and soft bands should be correlated.\n\nOverall, the {\\it ROSAT} PSPC data are qualitatively consistent with\nthis scenario. In the 1992 observation, the spectrum was harder at\nhigh flux and correlated variability was observed, while at low flux\nuncorrelated variability was found.  In the 1991 observation, when the\nflux was lower and the spectrum generally softer, only uncorrelated\nvariability was observed.  This scenario can also explain the lack of\nspectral variability seen in the {\\it ROSAT} All Sky Survey data\n(Molendi, Maccacaro and Schaeidt \\markcite{20} 1993) since at that\ntime the source was bright and power law emission may have dominated\nthe soft component emission.  Qualitatively and on short time scales\nthere are difficulties with this scenario.  In 1992, the hardest and\nsoftest bands decrease by 50\\% and 25\\% respectively overall, implying\nthe soft component must contribute the same percentages of the soft\nband flux in the high and low states respectively.  However, at high\nflux there is a dip in flux by 30\\% just after the start of the\nobservation, and no change in the softness ratio was observed.\nSimilarly, at low flux, there is an increase by 30\\% in the hard band\napproximately $6\\times 10^4\\,\\rm s$ after the start of the observation,\nbut no change in the soft band emission was observed.  In combination,\nthese two results cannot be explained if the black body flux is constant\nand no other parameters change.  Reprocessing, neglected so far, may\nbe able to explain large amplitude correlated variability at high\nflux.\n\nOther models cannot explain the observed spectral variability.\nNetzer, Turner \\& George (1994) \\markcite{24} showed that variability\nof the warm absorber in response to ionizing flux changes could not\nexplain the spectral variability found during the 1992 observation.\nIn general, if the soft component were dominated by reprocessing, the\nsoft X-ray variability would be expected to track the hard component\nvariability.  However, substantial spectral variability of the\nincident continuum could result in some spectral variability of the\nreprocessed component.  Changes in the accretion rate (Molendi \\&\nMaccacaro \\markcite{19} 1994) cannot explain the orbit-to-orbit\nspectral variability as the predicted time scales are much longer.\n\n\\subsection{The Change in the Warm Absorber}\n\nIn the {\\it ASCA} spectra, we found that there was no evidence that\nthe ionized absorption column density changed between the high state\nand the low state; however, we found that the ionization parameter\nchanged with 90\\% confidence.  This result is model dependent, as we\ncould not demonstrate that the optical depth of the oxygen edges\nchanged significantly.\n\nThe best fit ionization parameter $\\log(U)$ changed from $\\sim -0.85$\nin the low state to $\\sim -0.42$ in the high state, implying an\nincrease in flux of ionizing photons by a factor of 2.7.  It is\ninteresting that this is quite close to the implied change in flux by\na factor of 3.1 of the intrinsic power law at 0.7 keV. The observed\nphoton index variability implies a larger change in the flux of\nionizing photons, assuming the power law extends to low energies.  The\nspectral changes occurred over a time scale of several thousand\nseconds.  The recombination time scale for \\ion{O}{8} is about $2\n\\times 10^{11}T_e^{0.5} n^{-1}$ seconds (e.g.  Turner et al.\n\\markcite{33} 1993) or about 5.5 hours using the parameters assumed in\nthis model (or longer if the gas is rarer).  Thus the gas may not be\nin photoionization equilibrium in the high flux state, and the larger\npopulation of \\ion{O}{8} implied by the increase in the ionization\nparameter may result from ions which are directly stripped of an\nelectron by the increased number of photons with energy near 0.7 keV.\nFurther observations are necessary to determine the response of the\nionized material to a decrease in flux since a change in ionization\nwould be observed only if recombination had occurred.\n\nA change in the ionization correlated with an increase in flux has not\nbeen previously reported from {\\it ASCA} data.  An increase in column\ndensity and no change in ionization accompanied an increase in flux in\nMCG--6-30-15 (Fabian et al. \\markcite{8} 1994).  Explaining the\nspectral variability during an increase in flux by a change in warm\nabsorber properties in NGC~3227 required a decrease in ionization and\nan increase in column (Ptak et al. \\markcite{27} 1994). In another\nobservation of MCG--6-30-15, an increase in the optical depth of the\n\\ion{O}{8} edge during a flux decrease was interpreted as evidence for\nrecombination of \\ion{O}{9} (Otani \\markcite{26} 1995).\n\n\\subsection{Hard X-ray Emission from Narrow Line Seyfert 1s}\n\\vskip 1pc\n\nThe soft X-ray properties of narrow line Seyfert 1s are well studied\n(Boller, Brandt \\& Fink \\markcite{1} 1996) but few hard X-ray\nobservations have been reported. The {\\it ASCA} observation of Mrk~766\nrepresents one of the first observations of the hard emission from\nthese objects.\n\nWe found a hard power law with variable photon index in Mrk~766, and\nthe spectral variability which was not strictly flux correlated.\nSimilar photon index variability which was flux correlated has been\ndiscovered from NGC~4051 (Guainazzi et al. \\markcite{15} 1996), a\nSeyfert 1 galaxy which shares many properties with NLS1s.  In\ncontrast, a very steep spectrum with photon index $\\sim 2.6$, a\ndominate soft excess and no variability was observed from narrow-line\nSeyfert 1 RE~1034+39 (Pounds, Done \\& Osborne \\markcite{39} 1995).\nBecause these spectral and variability properties are similar to those\ncharacteristic of black hole candidates in the high state (e.g. Nowak\n\\markcite{41} 1991) it was postulated that RE~1034+39 represents a\nSeyfert 1 galaxy in the high state (Pounds, Done \\& Osborne\n\\markcite{39} 1995).\n\nThe marked differences between the hard X-ray properties of Mrk~766\nand RE~1034+39 are interesting.  Black hole candidates in the low\nstate are characterized by a flat hard X-ray power law and more rapid,\nlarger amplitude hard X-ray variability (e.g. Nowak \\markcite {41}\n1995).  These properties more closely resemble the observational\nresults from Mrk~766 and NGC~4051 than do the properties of black hole\ncandidates in the high state.  Further observations of NLS1s may find\nthat the spectral and variability properties fall into two classes:\nthose with steep hard X-ray spectra, dominant soft X-ray emission and\nlower amplitude short term hard X-ray variability, and those with flat\nhard X-ray spectra, less soft X-ray emission and rapid hard X-ray\nvariability.  There is perhaps already some evidence for such a\ndivision. While many NLS1s are very bright soft X-ray objects commonly\nfound in soft X-ray samples, relatively few have hard X-ray detections\nby HEAO-1 A2.\n\nFurther support for this scenario may come if repeated X-ray\nobservations discover that some objects have made the transition\nbetween two states.  This could have been what had occurred in objects\nobserved to have varied by factors of 10 or more between two {\\it\nROSAT} observations (e.g. Zwicky 159.034, Brandt, Pounds \\& Fink\n\\markcite{3} 1995; WPVS007, Grupe et al. \\markcite{14} 1995).\nWell-studied broad-line Seyfert 1 galaxies are not observed to undergo\nthis kind of transition.  The fact that many well-studied AGN are hard\nX-ray selected while NLS1s are clearly soft X-ray selected objects\nfurther supports this hypothesis.\n\nIn black hole candidates, it is widely believed that the high state\nis characterized by a relatively larger accretion rate compared with\nthe low state.  Thus the behavior of NLS1s may result from a\nrelatively larger accretion rate.  If the processes fueling AGNs are\ncommon for Seyferts, a relatively larger accretion rate would be\nobserved in systems with relatively small black hole masses.\n\n\\section{Conclusions}\n\nWe report analysis of {\\it ASCA} and {\\it ROSAT} observations of the\nnarrow-line Seyfert 1 galaxy Mrk~766.  In the {\\it ASCA} observation\nrapid variability with doubling time scale of order $\\sim 1000$\nseconds was observed, and dramatic spectral variability over as time\nperiod of less than $\\sim 10,000\\,\\rm s$ was discovered. Confined to a\nsingle event, during a 2--10 keV flux increase the spectrum above and\nbelow $\\sim 1$ keV softened and hardened respectively. The low and\nhigh flux spectra could be described with a model consisting of a\npower law, iron line, warm absorber and soft excess modeled as a black\nbody.  The spectral variability was a result of a highly significant\nincrease in the intrinsic power law index from $\\sim 1.6$ to $\\sim\n2.0$ with the pivot point at $\\sim 9$ keV, a model dependent increase\nin the ionization of the warm absorber, and a marginal decrease in the\nsoft excess component.  A $100 \\rm \\, eV$ equivalent width narrow iron\nline was detected in the high flux spectrum but not in the low flux\nspectrum, most likely because of poor statistics.  Variability on time\nscales as short as $\\sim 2400$ seconds was found in the {\\it ROSAT}\ndata.  Because the variability in the softest {\\it ROSAT} band, below\n0.4 keV, had relatively lower amplitude than the harder bands,\nspectral hardening during flux increases was detected on time scales\nas short as the orbital period of $ \\sim 6000 \\,\\rm s$.\n\nThe spectral index change, the disappearance of the soft component in\nthe {\\it ASCA} band and the confinement of the spectral variability to\na single event could be naturally explained in terms of non-thermal\nComptonization models.  We postulate that the index change occurred\nthrough a transition from a first order pair reprocessed spectrum to a\npair cascade spectrum brought about by a sudden increase in the\nLorentz factor of the injected relativistic electrons.  The first\norder pair reprocessed spectrum observed in the low state as a soft\nexcess disappeared in the high state cascade spectrum. Variations in\nthe hard compactness resulted in pure flux variability.  The measured\nincrease in the warm absorber ionization corresponds to the increase\nin flux near the oxygen edges resulting from the power law index\nchange.  The spectral variability in the {\\it ROSAT} data was most\nnaturally explained by a variable hard component and a nonvariable\nsoft component which dominated the softest band and may be primary\nemission from an accretion disk perhaps implying that reprocessing is\nrelatively less important in this object.\n\nThe flat and variable hard power law index observed in Mrk~766 is\nsimilar to that observed in NGC 4051 (Guainazzi et al. \\markcite{15}\n1996), a Seyfert 1 with many properties common to NLS1s, but contrasts\nmarkedly with the very steep hard X-ray index $\\Gamma \\sim 2.6$ found\nin NLS1 object RE~1034+39 (Pounds, Done \\& Osborne \\markcite{39} 1995).\nFurther hard X-ray observations of NLS1s using {\\it ASCA} are\nnecessary to clearly understand the hard X-ray properties of these\nsources.\n\n\\acknowledgements\n\nThis research has made use of data obtained through the High Energy\nAstrophysics Science Archive Research Center Online Service, provided\nby the NASA-Goddard Space Flight Center. The authors thank T. Kallman\nand P. Zycki for the use of their spectral table models.  KML\nacknowledges receipt of a National Research Council fellowship at\nNASA\/GSFC and a Japanese Science and Technology Agency fellowship at\nRIKEN and useful conversations with M. Cappi.  KML acknowledges\nsupport by a {\\it ROSAT} AO4 guest observer grant and an {\\it ASCA}\nAO1 guest observer grant.\n\n\\clearpage\n\n\\begin{deluxetable}{lrrrrr}\n\\tablewidth{0pc}\n\\tablenum{1}\n\\tablecaption{Observing Log}\n\\tablehead{\n\\colhead{Instrument}   & \\colhead{Observation} & \n\\colhead{Exposure}     & \\colhead{Total Counts} &\n\\colhead{Background}   \\\\\n\\colhead{}             & \\colhead{Date} &\n\\colhead{(s)}          & \\colhead{} &\n\\colhead{Count Rate}  }\n\n\\startdata\n{\\it ROSAT} PSPC & 15\/06\/91 & 15179\\tablenotemark{a} & 21804 & 0.081   \\nl\n{\\it ROSAT} PSPC & 21\/12\/92 & 16303\\tablenotemark{b} & 53865 & 0.056  \\nl\n{\\it ROSAT} PSPC & 17\/12\/93 & 3146 & 7309 & 0.033  \\nl\n{\\it ASCA} S0 & 18\/12\/93 & 32947 & 32823 & 0.030  \\nl\n\\hphantom{{\\it ASCA}} S1 & & 32735 & 26853 & 0.020 \\nl\n\\hphantom{{\\it ASCA}} G2 & & 35805 & 16991 & 0.033 \\nl\n\\hphantom{{\\it ASCA}} G3 & & 35808 & 19532 & 0.032 \\nl\n\\tablecomments{Column 4 is the total (not background subtracted) counts in the\nsource extraction region.  \nColumn 5 is the background rate scaled to the source\nextraction region.  }\n\\tablenotetext{a} {Exposure time after data selection to remove periods\nwhere the source was occulted by a rib.}\n\\tablenotetext{b} {Exposure time in the first 85,000 seconds of\nobservation.  The total observation time was 19870 seconds.}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\\begin{deluxetable}{lrrr}\n\\tablewidth{0pc}\n\\tablenum{2}\n\\tablecaption{PSPC Variability}\n\\tablehead{\n\\colhead{Band}   & \\colhead{Mean} & \n\\colhead{$\\chi^2_\\nu$}     & \\colhead{NVA}}\n\n\\startdata\n\n\\multicolumn{4}{l}{1991 Data (44 points):} \\nl\nTotal & 1.35 & 17.6 & 0.22 \\nl\n0.2--0.5 & 0.84 & 6.0 & 0.16 \\nl\n0.5--0.9 & 0.20 & 7.05 & 0.36 \\nl\n0.9--2.0 & 0.22 & 8.54 & 0.37 \\nl\n\\tableline\n\\multicolumn{4}{l}{1992 Data (41 points):} \\nl\nTotal & 3.06 & 77.5 & 0.26 \\nl\n0.2--0.5 & 1.63 & 22.7 & 0.19 \\nl\n0.5--0.9 & 0.58 & 24.9 & 0.34 \\nl\n0.9--2.0 & 0.67 & 35.1 & 0.38 \\nl\n\\tableline\n\\multicolumn{4}{l}{Quasi-simultaneous} \\nl\n\\multicolumn{4}{l}{Data (8 points):} \\nl\nTotal & 2.24 & 29.9 & 0.19 \\nl\n0.2--0.5 & 1.33 & 8.2 & 0.13 \\nl\n0.5--0.9 & 0.39 & 13.9 & 0.31 \\nl\n0.9--2.0 & 0.39 & 13.6 & 0.31 \\nl\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\\begin{deluxetable}{lcccc}\n\\footnotesize\n\\tablewidth{0pc}\n\\tablenum{3}\n\\tablecaption{{\\it ASCA} Spectral Fitting Results}\n\\tablehead{\n\\colhead{Parameter}   & \\multicolumn{2}{c}{Low State} &\n\\multicolumn{2}{c}{High State} \\\\\n\\colhead{} & \\colhead{SIS0:GIS2} & \n\\colhead{SIS1:GIS3} & \\colhead{SIS0:GIS2} &\n\\colhead{SIS1:GIS3}}\n\n\\startdata\n\\multicolumn{5}{l}{Power law model:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ &\n $\\rm Gal < 0.19$ & $\\rm Gal<0.20$ & $\\rm Gal<0.19$ & $\\rm Gal<0.192$ \\nl\nIndex & $1.65 \\pm 0.04$ & $1.64^{+0.04}_{-0.05}$ & $1.92 \\pm 0.02$ & \n$1.91^{+0.01}_{-0.02}$ \\nl\n$\\chi^2$\/d.o.f. & 422\/270 & 341\/254 &\n 819\/598 & 765\/597 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law plus black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $0.84^{+0.51}_{-0.48}$ & \n$0.75^{+0.53}_{-0.49}$\n& $0.95^{+0.22}_{-0.20}$ & $1.02^{+0.22}_{-0.20}$ \\nl\nIndex & $1.54^{+0.06}_{-0.07}$ & $1.54^{+0.08}_{-0.07}$ & $1.96 \\pm 0.03$ &\n$1.98^{+0.04}_{-0.03}$ \\nl\nSIS PL norm\\tablenotemark{a}  & $3.0^{+0.3}_{-0.2}$ & $3.0^{+0.3}_{-0.2}$ &\n$7.7 \\pm 0.03$ & $7.9 \\pm 0.03$ \\nl\nkT (eV)  & $88^{+8}_{-6}$ & $87^{+9}_{-7}$ & $77^{+5}_{-4}$ & $72 \\pm 5$ \\cr\nSIS bb norm\\tablenotemark{b} & $2.5^{+2.1}_{-1.2}$ & $2.3^{+2.1}_{-1.2}$ &\n$4.2^{+1.5}_{-1.3}$ & $5.4^{+2.0}_{-1.7}$ \\nl\n$\\chi^2$\/d.o.f. & 245\/268 & 208\/254 & 681\/596 & 647\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law plus edge:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal < 0.18$ & $\\rm Gal<0.20$ & $\\rm Gal<0.26$ &\n$\\rm Gal< 0.23$ \\nl\nIndex & $1.87 \\pm 0.04$ & $1.83^{+0.06}_{-0.05}$ & $2.01 \\pm 0.02$ &\n$2.01^{+0.03}_{-0.02}$ \\nl\nSIS PL norm\\tablenotemark{a}\n & $4.38 \\pm 0.18$ & $4.23^{+0.24}_{-0.23}$ & $8.06^{+0.17}_{-0.15}$\n& $8.06^{+0.23}_{-0.17}$ \\nl\nEdge Energy (keV) & $0.81 \\pm 0.02$ & $0.82^{+0.02}_{-0.03}$ &\n$0.78^{+0.01}_{-0.03}$ & $0.75 \\pm 0.02$ \\nl\n$\\tau$ & $1.01 \\pm 0.15$ & $0.89^{+0.21}_{-0.19}$ & $0.49^{+0.07}_{-0.06}$ &\n$0.53 \\pm 0.70$ \\nl\n$\\chi^2$\/d.o.f. & 344\/268 & 279\/254 & 651\/596 & 608\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law, black body and edge:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.28<0.91$ & $\\rm Gal<0.42<0.88$ &\n$0.43^{+0.18}_{-0.20}$ & $\\rm Gal<0.35<0.63$ \\nl\nIndex & $1.55^{+0.09}_{-0.05}$ & $1.57 \\pm 0.07$ & $1.99^{+0.02}_{-0.04}$ &\n$2.00^{+0.04}_{-0.03}$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.1^{+0.3}_{-0.2}$ & $3.2 \\pm 0.30$ &\n $7.9^{+0.2}_{-0.3}$ & $8.0^{+0.4}_{-0.3}$ \\nl\nEdge energy (keV) & $0.76^{+0.03}_{-0.04}$ & $0.74 \\pm 0.02$ &\n$0.74^{+0.03}_{-0.02}$ & $0.75^{+0.02}_{-0.03}$ \\nl\n$\\tau$ & $0.60^{+0.28}_{-0.25}$ & $0.77^{+0.28}_{-0.30}$ & \n$0.43^{+0.09}_{-0.10}$ & $0.46^{+0.12}_{-0.11}$ \\nl \n$kT$ (eV) & $117 \\pm 18$ & $121^{+21}_{-16}$ & $101^{+19}_{-13}$\n & $91^{+48}_{-35}$ \\nl\nSIS bb norm\\tablenotemark{b}\n & $0.95^{+1.19}_{-0.26}$ & $1.12^{+0.82}_{-0.44}$ & $0.81^{+0.56}_{-0.50}$\n& $0.54^{+1.04}_{-0.52}$ \\nl\n$\\chi^2$\/d.o.f. & 230\/266 & 191\/253 & 640\/594 & 605\/593 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law and 2 edges:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.22$ & $\\rm Gal<0.22$ & $1.1<2.8$ &\n$2.5<3.4$ \\nl\nIndex & $1.91^{+0.05}_{-0.04}$ & $1.90 \\pm 0.05$ & $2.02^{+0.05}_{-0.01}$ &\n$2.05 \\pm 0.04$ \\nl\nSIS PL norm\\tablenotemark{a} & $5.0 \\pm 0.20$ & $4.9 \\pm 0.30$ & \n$8.2^{+0.5}_{-0.2}$ & $8.5^{+0.5}_{-0.4}$ \\nl\nEdge energy (keV) & $0.78^{+0.01}_{-0.02}$ & $0.77 \\pm 0.03$ &\n$0.74^{+0.02}_{-0.04}$ & $0.73 \\pm 0.02$ \\nl\n$\\tau$ & $1.00^{+0.18}_{-0.15}$ & $1.01^{+0.23}_{-0.18}$ &\n $0.43^{+0.08}_{-0.19}$ & $0.51^{+0.09}_{-0.07}$ \\nl\nEdge energy (keV) & $1.19^{+0.05}_{-0.03}$ & $1.24^{+0.07}_{-0.14}$ &\n$0.94^{+0.06}_{-0.24}$ & $0.98^{+0.05}_{-0.04}$ \\nl\n$\\tau$ & $0.55 \\pm 0.11$ & $0.49^{+0.15}_{-0.11}$ & $0.16^{+0.15}_{-0.06}$ &\n$0.14^{+0.07}_{-0.06}$ \\nl\n$\\chi^2$\/d.o.f. & 254\/266 & 241\/252 & 631\/594 & 594\/593 \\nl\n\\tableline\n\\tablebreak\n\\multicolumn{5}{l}{Power law, 2 edges and black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $Gal<0.32<0.81$ & $\\rm Gal<0.70$ &\n$\\rm Gal<0.29<0.44$ & $\\rm Gal<0.49<0.52$ \\nl\nIndex & $1.64^{+0.10}_{-0.09}$ & $1.62 \\pm 0.09$ & $2.01 \\pm 0.04$ &\n$2.05 \\pm 0.04$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.5^{+0.5}_{-0.4}$ & $3.4^{+0.4}_{-0.3}$ &\n$8.2 \\pm 0.4$ & $8.5^{+0.5}_{-0.4}$ \\nl\nEdge energy (keV) & $0.76 \\pm 0.03$ & $0.75^{+0.02}_{-0.03}$ &\n$0.73^{+0.02}_{-0.04}$ & $0.73 \\pm 0.02$ \\nl\n$\\tau$ & $0.67^{+0.26}_{-0.27}$ & $0.84^{+0.37}_{-0.30}$ & \n$0.40^{+0.15}_{-0.16}$ & $0.51^{+0.08}_{-0.07}$ \\nl\n$kT$ (eV) & $115 \\pm 21$ & $134^{+31}_{-33}$ & $126^{+34}_{-29}$ & \n84\\tablenotemark{c} \\nl\nSIS bb norm\\tablenotemark{b} & $0.88^{+1.00}_{-0.37}$ &\n$0.69^{+1.04}_{-0.13}$ & $0.29^{+0.31}_{-0.26}$ & $0<0.08$ \\nl\nEdge energy (keV) & $1.20^{+0.07}_{-0.17}$ & 1.104\\tablenotemark{c}\n & $0.87^{+0.12}_{-0.06}$ & $0.98^{+0.05}_{-0.04}$ \\nl\n$\\tau$ & $0.18^{+0.16}_{-0.14}$ & $0<0.16<0.27$ & $0.18^{+0.14}_{-0.09}$ &\n$0.14 \\pm 0.07$ \\nl\n$\\chi^2$\/d.o.f.  & 226\/264 & 189\/250 & 629\/592 & 594\/591 \\nl\n\\tableline\n\\multicolumn{5}{l}{Warm Absorber:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.18$ & $\\rm Gal < 0.18$ &\n$\\rm Gal<0.23<0.30$ & $0.26^{+0.09}_{-0.08}$ \\nl\nIndex & $2.05^{+0.04}_{-0.07}$ & $1.98 \\pm 0.08$ & $2.12^{+0.05}_{-0.04}$ &\n$2.13^{+0.05}_{-0.04}$ \\nl\nSIS pl norm\\tablenotemark{a} & $6.4^{+0.5}_{-0.7}$ & $5.7^{+0.5}_{-0.3}$ &\n9$.7 \\pm 0.5$ & $9.9 \\pm 0.6$ \\nl\n$log(U)$ & $-0.17^{+0.08}_{-0.19}$ & $-0.28^{+0.06}_{-0.09}$ &\n$-0.38^{+0.05}_{-0.06}$ & $-0.43^{+0.08}_{-0.06}$ \\nl\n$log(N_w)$\\tablenotemark{c} & $22.27^{+0.09}_{-0.17}$ & $22.13 \\pm 0.09$ &\n$21.71 \\pm 0.07$ & $21.68^{+0.08}_{-0.06}$ \\nl\n$\\chi^2$\/d.o.f. & 294\/268 & 269\/254 & 634\/596 & 601\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Warm Absorber plus black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & Gal<0.48<0.94 & $\\rm Gal<0.36<0.79$ &\n$\\rm Gal<0.29<0.42$ & $\\rm Gal<0.23<0.40$ \\nl\nIndex & $1.70^{+0.13}_{-0.11}$ & $1.70^{+0.14}_{-0.10}$ &\n $2.12^{+0.05}_{-0.07}$ & $2.10^{+0.07}_{-0.06}$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.9^{+1.0}_{-0.6}$ & $4.0^{+1.1}_{-0.6}$ &\n$9.8^{+0.9}_{-0.8}$ & $9.5^{+1.0}_{-0.6}$ \\nl\n$log(U)$ & $-0.78^{+0.21}_{-0.32}$ & $-0.79^{+0.25}_{-0.26}$ & \n$-0.41^{+0.08}_{-0.07}$ & $-0.42^{+0.06}_{-0.05}$ \\nl\n$log(N_w)^{d}$ & $21.78^{+0.36}_{-0.39}$ & $21.84^{+0.40}_{-0.36}$ & \n$21.82^{+0.13}_{-0.20}$ & $21.83 \\pm 0.11$ \\nl\n$kT$ (eV) & $119^{+47}_{-18}$ & $131^{+55}_{-24}$ & $184^{+76}_{-103}$ &\n$230^{+105}_{-57}$ \\nl\nSIS bb norm\\tablenotemark{b}\n & $1.52^{+1.34}_{-0.66}$ & $1.37^{+1.70}_{-0.48}$ & $0.49^{+0.71}_{-0.46}$\n& $0.49^{+0.65}_{-0.44}$ \\nl\n$\\chi^2$\/d.o.f. & 234\/266 & 194\/253 & 630\/594 & 596\/593 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power Law, two edges and Reflection:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal < 0.39$ & $\\rm Gal < 0.50$ &\n$\\rm Gal<0.23<0.37$ & $\\rm Gal<0.28<0.44$ \\nl\nIndex & $2.05^{+0.12}_{-0.06}$ & $2.04^{+0.17}_{-0.06}$ & \n$2.06^{+0.08}_{-0.05}$ & $2.07^{+0.09}_{-0.06}$ \\nl\nSIS PL norm\\tablenotemark{a} & $4.87^{+0.52}_{-0.22}$ & $4.75^{+0.83}_{-0.26}$ &\n$8.40^{+0.61}_{-0.36}$ & $8.65^{+0.58}_{-0.51}$ \\nl\nSIS Refl norm\\tablenotemark{a}\n & $2.73^{+1.83}_{-1.10}$ & $2.56^{+2.42}_{-0.94}$ & $0.21<1.24$ &\n$0.26<1.35$ \\nl\nRefl. Ratio & $5.5^{+2.5}_{-2.1}$ & $8.3^{+6.1}_{-2.6}$ & $0.4<1.5$ &\n$0.3<1.5$ \\nl\nEdge Energy (keV) & $1.18^{+0.05}_{-0.1}$ & $1.11^{+0.05}_{-0.06}$ &\n$0.94^{+0.07}_{-0.11}$ & $0.99 \\pm 0.04$  \\nl\n$\\tau$ & $0.44^{+0.08}_{-0.12}$ & $0.38^{+0.13}_{-0.12}$ & \n$0.18^{+0.06}_{-0.08}$ & $0.15 \\pm 0.07$ \\nl\nEdge Energy (keV) & $0.77 \\pm 0.02$ & $0.75 \\pm 0.02$ &\n$0.73 \\pm 0.02$ & $0.73^{+0.02}_{-0.01}$ \\nl\n$\\tau$ & $0.98 \\pm 0.17$ & $0.90^{+0.18}_{-0.17}$ & $0.44^{+0.10}_{-0.20}$ &\n$0.53\\pm 0.09$ \\nl\n$\\chi^2$\/d.o.f. & 230\/263 & 201\/250 & 631\/593 & 594\/590 \\nl\n\\tablecomments{The value ``Gal'' for the absorption refers to the spectral\nfit lower limit set to the Galactic value, $1.77 \\times 10^{20} \\rm \\,\ncm^{-2}$ (Elvis, Wilkes \\& Lockman \\markcite{6} 1989).}\n\\tablenotetext{a}{$\\times\n10^{-3} \\rm photons\\,keV^{-1}cm^{-2}s^{-1}$}\n\\tablenotetext{b}{$\\times 10^{-4} L_{39}\/{D_{10}^2}$, where \n$L_{39}$ is the source\nluminosity in $10^{39}\\rm erg\\,s^{-1}$ and $D_{10}$ is the distance to\nthe source in $10^{10} \\rm kpc$}\n\\tablenotetext{c}{Unconstrained}\n\\tablenotetext{d}{log of the ionized column in units of $\\rm cm^{-2}$.}\n\\enddata\n\n\\end{deluxetable}\n\\clearpage\n\n\n\\begin{deluxetable}{lcc}\n\\tablewidth{0pc}\n\\tablenum{4}\n\\tablecaption{Iron Line Fitting Results}\n\\tablehead{\n\\colhead{Parameter}   & \\colhead{Low State} & \n\\colhead{High State}}\n\n\\startdata\n\\multicolumn{3}{l}{Power law plus narrow line:} \\nl\nIndex & $1.57 \\pm 0.07$ & $2.00^{+0.03}_{-0.04}$ \\nl\nLine Flux\\tablenotemark{a} & $1.8<3.0$ & $2.3^{+1.0}_{-1.2}$ \\nl\nLine Eq. Width (eV) & $100<170$ & $110^{+40}_{-55}$ \\nl\n$\\chi^2$\/d.o.f. & 204\/252 & 793\/758 \\nl\n\\tableline\n\\multicolumn{3}{l}{Power law, narrow line and reflection:} \\nl\nIndex & $1.66 \\pm 0.07$ & $2.08^{+0.04}_{-0.03}$ \\nl\nLine Flux\\tablenotemark{a}& $1.1<2.6$ & $1.8^{+0.9}_{-1.0}$ \\nl\nLine Eq. Width (eV) & $35<84$ & $47 \\pm 25$ \\nl\n$\\chi^2$\/d.o.f. & 204\/252 & 793\/758 \\nl\n\\tablenotetext{a}{$\\times 10^{-5} \\rm photons\\,cm^{-2}s^{-1}$ in the\nline.}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\n\\begin{deluxetable}{lcccc}\n\\footnotesize\n\\tablewidth{0pc}\n\\tablenum{5}\n\\tablecaption{Combined Spectral Fits}\n\\tablehead{\n\\colhead{Parameter}   & \\multicolumn{2}{c}{SIS0:GIS2}\n   & \\multicolumn{2}{c}{SIS1:GIS3} \\\\\n\\colhead{} & \\colhead{Low State} &\n\\colhead{High State} & \\colhead{Low State} &\n\\colhead{High State}}\n\n\\startdata\n\\multicolumn{5}{l}{Two Edge Model: } \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & \\multicolumn{2}{c}{$\\rm Gal<0.27<0.45$} &\n\\multicolumn{2}{c}{$\\rm Gal<0.25<0.37$} \\nl\nIndex & $1.58 \\pm 0.08$ & $2.02 \\pm 0.04$ & $1.60^{+0.08}_{-0.07}$ &\n$2.05 \\pm 0.04$ \\nl \nSIS PL norm\\tablenotemark{a} & $3.2 \\pm 0.30$ & $8.3^{+0.4}_{-0.3}$ \n& $3.2 \\pm 0.3$ & $8.6 \\pm 0.4$ \\nl\nkT (eV) & \\multicolumn{2}{c}{$123^{+22}_{-17}$} &\n \\multicolumn{2}{c}{$133^{+26}_{-21}$} \\nl\nSIS BB norm\\tablenotemark{b} & 0$.89^{+0.34}_{-0.22}$ & $0.19<0.61$ & \n$0.84^{+0.24}_{-0.16}$ & $0<0.23$ \\nl\nEdge Energy (keV) & \\multicolumn{2}{c}{$0.73^{+0.03}_{-0.01}$} &\n\\multicolumn{2}{c}{$0.74^{+0.01}_{-0.02}$} \\nl\n$\\tau$ & $0.66^{+0.28}_{-0.25}$ & $0.47^{+0.08}_{-0.09}$ & \n$0.83^{+0.19}_{-0.17}$ & $0.52^{+0.07}_{-0.08}$ \\nl\nEdge Energy (keV) & \\multicolumn{2}{c}{$0.97^{+0.05}_{-0.16}$} &\n\\multicolumn{2}{c}{ $0.99\\pm 0.04$ } \\nl\n$\\tau$ & $0.11<0.30$ & $0.15^{+0.06}_{-0.08}$ & $0.13<0.30$ & \n$0.14^{+0.07}_{-0.06}$\\nl\n$\\chi^2$\/d.o.f & \\multicolumn{2}{c}{860.1\/860} &\n\\multicolumn{2}{c}{790.7\/845 } \\nl \n\\tableline \n\\multicolumn{5}{l}{Warm Absorber Model: } \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & \\multicolumn{2}{c}{$0.33^{+0.16}_{-0.15}$} &\n\\multicolumn{2}{c}{$0.30^{+0.17}_{-0.10}$} \\nl\nIndex & $1.66^{+0.07}_{-0.06}$ & $2.12^{+0.05}_{-0.04}$ &\n$1.67^{+0.07}_{-0.06}$ & $2.14^{+0.05}_{-0.04}$ \\nl\nSIS PL norm\\tablenotemark{a}\n & $3.7^{+0.2}_{-0.3}$ & $9.8 \\pm 0.6$ & $3.6^{+0.3}_{-0.2}$ &\n$10.0 \\pm 0.6$ \\nl \nkT (eV)  & \\multicolumn{2}{c}{$117^{+13}_{-14}$} & \n\\multicolumn{2}{c}{$121^{+12}_{-14}$} \\nl\nSIS BB norm\\tablenotemark{b}\n & $1.34^{+0.43}_{-0.38}$ & $0.42<0.97$ & $1.20^{+0.49}_{-0.28}$ &\n$0.02<0.52$ \\nl\n$log(N_w)$\\tablenotemark{c} & \\multicolumn{2}{c}{$21.68^{+0.04}_{-0.08}$} \n& \\multicolumn{2}{c}{$21.69 \\pm 0.08$}\n\\nl\n$log(U)$ & $-0.85^{+0.16}_{-0.15}$ & $-0.42 \\pm 0.08$  &\n$-0.87^{+0.16}_{-0.15}$ & $-0.42^{+0.07}_{-0.09}$ \\nl\n$\\chi^2$\/d.o.f & \\multicolumn{2}{c}{867.1\/863} & \\multicolumn{2}{c}{801.8\/848} \\nl\n\\tablecomments{The value ``Gal'' for the absorption refers to the spectral\nfit lower limit set to the Galactic value, $1.77 \\times 10^{20} \\rm \\,\ncm^{-2}$ (Elvis, Wilkes \\& Lockman \\markcite{6} 1989).}\n\\tablenotetext{a} {$\\times 10^{-3} \\rm\nphotons\\,keV^{-1}cm^{-2}s^{-1}$}\n\\tablenotetext{b} {$\\times 10^{-4} L_{39}\/{D_{10}^2}$, where $L_{39}$\n is the source\nluminosity in $10^{39}\\rm erg\\,s^{-1}$ and $D_{10}$ is the distance to\nthe source in $10^{10} \\rm kpc$.}\n\\tablenotetext{c}{log of the ionized column in units of $ \\rm\ncm^{-2}$}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\nThe coupling between density fluctuations of different wavelengths is one of the\nmost important topics in the study of large-scale structure \\cite{Bernardeau:2001qr}.\nThese couplings can be imprinted in the inflationary initial conditions or develop \nthrough  gravitational evolution.  In the latter class, a long-wavelength density mode\naffects the evolution of all short-wavelength modes that are embedded in it leading\nto changes in the power spectrum \\cite{Takada:2013bfn,Li:2014jra,Chiang:2014oga,Chiang:2015eza}\nand the dark matter halo abundance which gives rise to halo bias \\cite{Mo:1995cs,Seljak:2012tp}.\nIn the limit of a large separation in these scales, one can use the ``separate universe''\n(SU) approach to describe these and other effects through a change in the background\ndensity within which small scale structure evolves \n\\cite{1993MNRAS.262..717B,Baldauf:2011bh,Sherwin:2012nh}.\n\n\nThe SU approach is not only conceptually straightforward to understand but can also\nbe readily implemented in cosmological simulations, arbitrarily deep into the nonlinear\nregime where perturbation theory breaks down \\cite{Sirko:2005uz,Gnedin:2011kj,Li:2014sga,Wagner:2014aka}.\nIn particular, SU simulations have enabled studies of the squeezed-limit $n$-point\ncorrelation functions \\cite{Wagner:2015gva} and their impact on the power spectrum\ncovariance \\cite{Li:2014sga}, the halo bias \\cite{Li:2015jsz,Lazeyras:2015lgp,Baldauf:2015vio},\nand the Lyman-$\\alpha$ forest \\cite{McDonald:2001fe,Cieplak:2015kra}. Since the whole\ntime evolution of the long-wavelength mode is properly captured, as opposed to just\na single epoch such as the time of observation, temporally nonlocal effects on small-scale\nobservables such as the nonlinear power spectrum \\cite{Ma:2006zk} and halo bias\n\\cite{Senatore:2014eva,LoVerde:2014pxa} are correctly modeled.\n\n\nPrevious studies have focused on SU simulations in the $\\Lambda$CDM cosmology,\nwhere only matter clusters at low redshift. If the system contains additional\nclustering components such as dynamic dark energy or massive neutrinos, then\none has to be careful when applying the separate universe principle. Specifically,\nthe separate universe construction is only strictly true if long-wavelength\nperturbations evolve under gravitational forces alone and not internal stress\ngradients \\cite{Dai:2015jaa,Hu:2016ssz}. This means that a SU description would\nseem to require that the long-wavelength mode be larger than the Jeans or free\nstreaming scale of the system. On the other hand, if the impact on small-scale\nstructure of these extra components is only gravitational, then it can be correctly\nmodeled by matching the local expansion rate to an SU Hubble rate in a ``fake''\nSU approach which implicitly requires fictitious energy density components \\cite{Hu:2016ssz}.\n\n\nIn this work, we implement and test this multi-component SU method in simulations\nwith quintessence dark energy. In particular, the growth of long-wavelength matter\nfluctuations above or below the Jeans scale of quintessence differs due to clustering\nof the dark energy. As a result, the SU expansion history depends on the scale of\nthe long-wavelength matter fluctuation as does the response of small-scale observables\nsuch as the power spectrum and halo mass function. The latter implies that halo bias\nitself will become scale dependent.\n\n\nThe rest of the paper is organized as follows.\nIn \\refsec{theory}, we describe the mapping of perturbations in the quintessence model\nonto the SU background above and below the Jeans scale.\nIn \\refsec{sims}, we implement  the SU approach in quintessence simulations.\nWe present the results of SU simulations in \\refsec{pk} and \\refsec{bias} for \nthe power spectrum response and the  halo bias respectively.\nWe discuss these results in \\refsec{discussion}.\nIn \\refapp{bias_model}, we compare our results to the predictions of scale-dependent halo bias models\nin the recent literature.\n\n\nThroughout the paper, we  adopt a spatially flat cosmology with a Hubble constant\n$h=0.7$, matter density $\\Omega_m=0.3$, quintessence energy density $\\Omega_Q=0.7$,\nquintessence equation of state $w_Q=-0.5$, and an initial curvature power spectrum\nwith scale-invariant tilt $n_s=1$ and amplitude which sets $\\sigma_8=1$ today.\nThese parameters are chosen to highlight the scale dependence of quintessence\nrather than for observational viability.\n\n\n\\section{Quintessential Separate Universe}\n\\label{sec:theory}\n\n\nFollowing Ref.~\\cite{Hu:2016ssz}, we review here the construction of the separate universe\nfor the case where components other than the cold dark matter possess Jeans scales. In\n\\refsec{expansion}, we show that the influence of these components is captured by a modified\nexpansion history that is defined by the growth history of the large-scale matter density\nfluctuation. We apply this construction to quintessence dark energy models in \\refsec{quintessence}.\n\n\\subsection{Expansion History}\n\\label{sec:expansion}\n\nA observer sitting within a long-wavelength matter fluctuation $\\delta_m$\nwould measure the {\\it local} mean matter density as\n\\begin{equation}\n \\bar{\\rho}_{mW}(a)=\\bar{\\rho}_m(a)[1+\\delta_m(a)] \\,,\n \\label{eq:rhoW}\n\\end{equation}\nwhere $W$ denotes a windowed average across a scale much smaller than the\nlong-wavelength mode. In the SU picture, the local mean evolves as if the\nobserver were in a SU whose scale factor \n\\begin{equation}\n a_W=\\frac{a}{(1+\\delta_m)^{1\/3}}\\approx a\\left(1-\\frac{\\delta_m}{3}\\right) \\,,\n \\label{eq:aW}\n\\end{equation}\nso that $\\bar{\\rho}_{mW} \\propto a_W^{-3}$.\nNote that at early times\n\\begin{equation}\n \\lim_{t\\to 0}\\delta_m\\to 0 \\,, ~~ \\lim_{t\\to 0}a_W\\to a \\,,\n\\end{equation}\nand the physical conditions of the local and global cosmology coincide. We\nhave implicitly assumed that there is a universal time coordinate between\nthe two and so in the relativistic limit $\\delta_m$ is specifically the\nsynchronous gauge density perturbation \\cite{Hu:2016ssz}.\n\n\nNotice that the SU construction requires only $\\delta_m(a)$ itself, not the\nevolution of any other density component in the universe. The other components \ndetermine the evolution of $\\delta_m(a)$, but they do not enter into $a_W$\nexplicitly. If these components only influence small-scale observables through\ntheir impact on $\\delta_m(a)$, their effects can be characterized by $a_W$\nand the {\\it local} Hubble expansion\n\\begin{equation}\n H_W=\\frac{\\dot{a}_W}{a_W}=H-\\frac13\\dot{\\delta}_m=H\\left(1-\\frac13\\delta'_m\\right) \\,,\n\\label{eq:H_W}\n\\end{equation}\nwhere $'\\equiv d\/d\\ln a$. This expansion history does not even need to be\ngiven by a SU Friedmann equation involving the local energy densities and\ncurvature \\cite{Gnedin:2011kj,Hu:2016ssz}. With $a_W$ and $H_W$ alone, we\ncan model the small-scale observables using $N$-body simulations with this\nSU expansion rate.\n\n\nThis construction includes cases where the other components experience\nnon-gravitational forces which define their Jeans scales. In these cases,\nthe growth history of $\\delta_m(a)$ depends on scale. Since the SU expansion\nhistory depends on the whole growth history, regions of different sizes that\nshare a common $\\delta_m$ at a fixed $a$ will produce different responses\nin the small-scale observables. In other words, these observables cannot be\ndescribed solely by the change in the local density at the time of observation\nalone. For example, as we shall see in \\refsec{bias}, the response of the\ndark matter halo abundance to $\\delta_m(a)$ leads to a halo bias that violates\nthe local bias expectation of scale independence in the linear regime.\n\n\n\\subsection{Quintessence}\n\\label{sec:quintessence}\n\n\nQuintessence or scalar field dark energy models provide a simple arena to\nexplore the response of small-scale observables to long-wavelength fluctuations,\nin particular their amplitude, scale, and growth history. The construction\nof the SU with quintessence perturbations has been extensively discussed in\nRef.~\\cite{Hu:2016ssz}. Here we only summarize the results that are related\nto simulating observable responses above and below the quintessence Jeans scale.\n\n\nThe sound speed of quintessence $c_Q$ sets the sound horizon or Jeans scale\n$r_J \\sim c_Q \/a H$ across which its influence on the evolution of $\\delta_m$\ndiffers. If $\\delta_m$ has a wavelength smaller than $r_J$, the quintessence\nperturbation is Jeans stable and becomes negligible in comparison. Thus the\nmatter fluctuations evolve under\n\\begin{equation}\n \\del{\\downarrow}''+\\left(2+\\frac{H'}{H}\\right)\\del{\\downarrow}'\n =\\frac32\\frac{H_0^2}{H^2}\\frac{\\Omega_m}{a^3}\\del{\\downarrow}\\,,\n\\label{eq:dm_subJ}\n\\end{equation}\nwhere $\\delta_m=\\del{\\downarrow}$ and the down arrow in the subscript denotes\nthe sub-Jeans case. On the other hand, if $\\delta_m$ has a wavelength larger\nthan $r_J$, then the quintessence perturbation $\\delta_Q$ has an impact on\n$\\delta_m$. Assuming that all fluctuations arise from initial curvature\nfluctuations and the sound speed of quintessence is much smaller than speed\nof light, we have for the two-component system \\cite{Hu:2016ssz}\n\\ba\n \\delta'_Q-3w_Q\\delta_Q\\:&=(1+w_Q)\\del{\\uparrow}' \\,, \\nonumber\\\\\n \\del{\\uparrow}'' +\\left(2+\\frac{H'}{H}\\right)\\del{\\uparrow}'\\:&=\n \\frac32\\frac{H_0^2}{H^2}\\left[\\frac{\\Omega_m \\del{\\uparrow}}{a^3}\n +\\frac{\\Omega_Q \\delta_Q}{a^{3(1+w_Q)}}\\right] \\,,\n\\label{eq:dm_supJ}\n\\ea\nwhere $\\delta_m =\\del{\\uparrow}$ and the up arrow  in the subscript denotes\nthe super-Jeans case. For simplicity we have also taken the quintessence\nequation of state parameter $w_Q=\\bar p_Q\/\\bar \\rho_Q$ to be a constant.\nWith the assumed curvature initial conditions, the initial conditions for\nthe fields are set by taking $\\delta_m=\\del{\\uparrow}=\\del{\\downarrow}$\nand $\\delta_Q$ are all proportional to $a$ in the matter dominated limit.\n\n\nIn \\refFig{dm} we plot $\\delta_m\/\\delta_{m0}$ as a function of the global\nscale factor, where $\\delta_{m0}=\\delta_m(a=1)$ is the present-day overdensity.\nThe red solid and blue dashed lines show the sub-Jeans and super-Jeans SUs.\nNormalized to the same $\\delta_{m0}$, the super-Jeans SU is always closer\nto the global universe ($\\delta_m=0$) in the past than the sub-Jeans SU in\nits expansion history. This implies that the response of the small-scale\nobservables such as the power spectrum and halo abundance should be smaller\nin the super-Jeans than the sub-Jeans SU.\n \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dm.pdf}\n\\caption{(Top) Scale-dependent growth in $\\delta_m$ as a function of the scale\nfactor for super-Jeans (red solid) and sub-Jeans (blue dashed) long-wavelength\nmodes. (Bottom) The ratio of $\\delta_m$ in super-Jeans to sub-Jeans cases. When\ncharacterized as a separate universe, the former is closer to the global universe\nthan the latter in the past for the same density fluctuation today.}\n\\label{fig:dm}\n\\end{figure}\n\n\nFinally for setting up simulations in the next section it is useful to define\nthe linear growth of short-wavelength structure in the SU. If the small-scale\nmatter fluctuations of interest are well within $r_J$, the growth function $D_W$\nis simply \\refeq{dm_subJ} with the SU expansion history\n\\begin{equation}\n \\frac{d^2D_W}{d\\ln a_W^2}+\\left(2+\\frac{d\\ln H_W}{d\\ln a_W}\\right)\\frac{dD_W}{d\\ln a_W}\n =\\frac32\\frac{H_{0W}^2}{H_W^2}\\frac{\\Omega_{mW}}{a_W^3}D_W \\,.\n\\label{eq:DW}\n\\end{equation}\nNote that \n\\begin{equation}\n \\Omega_{mW} H_{0W}^2 = \\Omega_m H_0^2 \\,,\n\\end{equation}\nand so only the SU expansion rate $H_W$ from \\refeq{H_W} is required to solve\nfor $D_W(a_W)$. As we shall see next, we can generalize this statement to nonlinear\nobservables with SU $N$-body simulations.\n\n\n\\section{Separate Universe Simulations}\n\\label{sec:sims}\n\n\nThe growth history of the long-wavelength matter fluctuation $\\delta_m(a)$ in\nthe global universe alone sets the expansion history in the separate universe.\nAll effects from long-wavelength fluctuations in other species are incorporated\ninto its growth history. In the quintessence model, within the Jeans scale dark\nenergy perturbations can be ignored and so the response of small scale observables\ncan be calibrated using $N$-body simulations with just this change in the expansion\nhistory.\n   \n\nUnlike the SU technique in $\\Lambda$CDM, the change of cosmological parameters \nand their correspondence with real energy densities and curvature becomes non-trivial\nfor the quintessence model \\cite{Hu:2016ssz} whereas the direct change in the\nexpansion rate $H_W$ remains simply determined by $\\delta_m(a)$. Thus, while\nsome steps are similar to the $\\Lambda$CDM SU techniques for running and analyzing\nsimulations (see e.g. \\cite{Li:2014sga,Wagner:2014aka,Li:2015jsz,Lazeyras:2015lgp}),\nthere are some major differences in performing the SU simulations with quintessence\nwhich we now describe.\n\n\nLet us start with setting the initial conditions for the simulations. Recall that \nat high redshift the separate and global universes are identical in their physical\ndescription. To achieve this, we first compute the linear power spectrum with the\nglobal cosmology at $z=0$ using CAMB \\cite{Lewis:1999bs,Howlett:2012mh}. We then\nrescale this power spectrum to the initial redshift of the simulations $a_{Wi}=0.02$ as\n\\begin{equation}\n P_W(k,a_{Wi})=P(k,a_0)\\left[\\frac{D_W(a_{Wi})}{D(a_0)}\\right]^2 \\,,\n\\end{equation}\nwhere $D$ is the linear growth in the global universe and $D_W$ is the linear\ngrowth of the SU following \\refEq{DW}. The growth functions are normalized in\nthe matter dominated epoch as\n\\begin{equation}\n \\lim_{a \\rightarrow 0} D(a)=a, \\quad \\lim_{a_W \\rightarrow 0} D(a_W) = a_W \\,.\n\\label{eq:D_ic}\n\\end{equation}\nNote that $D_W$ in sub-Jeans and super-Jeans SUs are different, as they have different\nexpansion histories. Another subtlety is that the change in the expansion rate of the\nSU makes the traditional unit of comoving $[h\\,{\\rm Mpc}^{-1}]$ inconvenient. Throughout this paper\nwe avoid this confusion by using units of comoving $[{\\rm Mpc}]$ and convert for code\npurposes as necessary. Given the different scale factors $a$ and $a_W$, the correspondence\nbetween comoving wavenumber and physical wavenumber in the global universe differ. Since\nthis represents a simple dilation of scales, we can account for it in the interpretation\nof observable responses rather than in the simulations directly \\cite{Li:2014sga}.\n \n\nThe initial conditions are then set up using realizations of Gaussian random\nfields for the primordial fluctuations and evolved to $a_{Wi}$ using second-order\nLagrangian perturbation theory (2LPT) \\cite{Crocce:2006ve}. Usual 2LPT codes,\nsuch as the publicly available 2LPTIC \\cite{2lptic}, compute the linear growth and\ngrowth rate $f_W=d\\ln D_W\/d\\ln a_W$ at $a_{Wi}$ from the cosmological parameters.\nWe modify the pipeline such that $D_W$ and $f_W$ from the numerical solution of\n\\refEq{DW} determine the initial positions and velocities of the particles.\n\n\nWe use Gadget-2 \\cite{Springel:2005mi} to carry out the simulations. Standard\nGadget-2 computes the Hubble expansion as a function of the scale factor using\nthe input cosmological parameters. Instead of finding the corresponding cosmological\nparameters, we first compute $H_W$ as a function of $a_W$ with \\refEq{H_W} and\n\\refEqs{dm_subJ}{dm_supJ}, pass the table $(a_W,H_W)$ to the code, and then\ninterpolate the value of $H_W(a_W)$ when necessary\\footnote{Specifically, we\nonly need to modify \\texttt{driftfac.c} and \\texttt{timestep.c}. Also since\nGadget-2 checks the consistency of the input parameters, we provide the SU\n$\\Omega_{mW}$ as well as $h_W$, and $L_W$ where $L_W$ is the box size of the\nsimulations}. We have verified that the SU results are in excellent agreement\nwith those of the standard 2LPTIC Gadget-2 pipeline in $\\Lambda$CDM where the\nSU is implemented by varying cosmological parameters.\n\n\nFollowing the procedures in Ref.~\\cite{Lazeyras:2015lgp}, we identify halos with\nthe Amiga Halo Finder \\cite{Knollmann:2009pb,Gill:2004km}, which is based on the\nspherical overdensity algorithm. The key quantity of the spherical overdensity\nalgorithm is the density threshold, and we set it to be $\\Delta=200$ in the global\nuniverse. To match halos identified in the global cosmology, the threshold relative\nto the mean in the SU needs to be rescaled as \\cite{Li:2015jsz,Lazeyras:2015lgp}\n\\begin{equation}\n \\Delta_W=\\frac{\\Delta}{1+\\delta_m(t)}\\approx\\Delta[1-\\delta_m(t)] \\,.\n\\end{equation}\nIn other words, in the overdense (underdense) universe the threshold becomes smaller\n(larger) due to the background fluctuations. From each simulation we obtain one halo\ncatalog, and we consider only halos with more than 400 particles. We also neglect\nsub-halos for simplicity.\n\n\nIn this paper, we perform both the sub-Jeans and super-Jeans SU simulations with\n$\\del{\\uparrow\\downarrow 0}=\\del{\\uparrow\\downarrow}(a=1)=\\pm0.01$, totaling 4\nsimulations per set. For each of the 20 sets, we fix their initial phases so that\nwhen we take the difference of the observables between overdense and underdense\nSU simulations a large amount of noise due to sample variance is removed. We also\nrun 40 simulations of the global $\\delta_m=0$ universe in order to characterize\nthe clustering bias for comparison in \\refsec{clusteringbias}. The first 20 have\nthe same initial phases as their SU counterparts. For each of these sets we take\na comoving box size $L=1000\\,$Mpc and number of particles $N_p=1024^3$, denoted\nas small-box.\n\n\nWe also run 20 simulations with $L=2800$ Mpc and $N_p=1024^3$ particles in the global\nuniverse, denoted as big-box simulations. These big-box simulations are used to measure\nthe position-dependent power spectrum \\cite{Chiang:2014oga} for comparison with the\npower spectrum response of the sub-Jeans simulations. The details of the simulations\nare summarized in \\reftab{sims}.\n\n\n\\begin{table}[h]\n \\begin{tabular}{c c c c c c c}\n  \\hline\n  type & SU & $L$ [Mpc] & $N_p$ & $\\del{m0}$ & $N_{\\rm sets}$\\\\\n  \\hline\n  small-box & $\\uparrow\\downarrow$  &1000 & $1024^3$ & $\\pm0.01$ & 20 \\\\\n  small-box &  no &1000 & $1024^3$ & 0 & 40 \\\\\n  big-box &  no&  2800 & $1024^3$ & 0 & 20 \\\\\n  \\hline\n \\end{tabular}\n \\caption{Summary of the simulations.}\n\\label{tab:sims}\n\\end{table}\n\n\n\\section{Power Spectrum Response}\n\\label{sec:pk}\n\n\nIn this section, we calibrate the responses in the locally measured power spectrum\nto a long-wavelength mode above and below the Jeans scale. In \\refsec{resp} we extract\nthese responses from the SU simulations and show that they are scale dependent and\nsmaller for modes above the Jeans scale than below. We test these responses against\npredictions from perturbation theory in \\refsec{pt} and the local, position-dependent,\npower spectrum from the big-box simulations with long-wavelength sub-Jeans scale modes\nin \\refsec{ibn}. The good agreement implies that the SU simulation technique provides\naccurate predictions for these small scale observables without the need for direct\nsimulations of quintessence clustering.\n\n\n\\subsection{Separate Universe Calibration}\n\\label{sec:resp}\n\n\nIn the presence of a long-wavelength density fluctuation $\\delta_m$, the power spectrum\nobserved locally will differ from the global average. We can characterize the fractional\nchange in the local power spectrum  as a ``response'' $R_{\\rm tot}$ to  $\\delta_m$\n\\begin{equation}\n \\frac{\\Delta P}{P} \\approx \\frac{d \\ln P}{d\\delta_m} \\delta_m \\equiv R_{\\rm tot} \\delta_m \\,.\n\\end{equation}\nSince to the leading order $R_{\\rm tot}$ is independent of $\\delta_m$, it can be calibrated\nusing the SU simulations once and for all rather than with simulations that follow the\ndynamics of individual long-wavelength modes. This is especially advantageous for quintessence,\nwhere super-Jeans modes require simulations with quintessence clustering.\n\n\nThis effect can be observed in a local sample of our universe by dividing it into subvolumes\nand measuring the correlation between the local power spectra and the subvolume mean overdensities,\nwhich is known as the position-dependent power spectrum \\cite{Chiang:2014oga,Chiang:2015eza}.\nEven if only the undivided volume is employed, the coherent change in the local power spectrum\n$\\Delta P(k)$ due to wavelengths larger than the sample induces a ``super-sample'' covariance\nbetween measurements of different $k$ modes \\cite{Takada:2013bfn,Li:2014sga,Li:2014jra}.\n\n\nIn practice, the calibration of the total response with SU simulations involves three pieces:\ngrowth, dilation, and reference-density \\cite{Li:2014sga}\n\\begin{equation}\n R_{\\rm tot}  = R_{\\rm growth} + R_{\\rm dilation} + R_{\\bar \\rho} \\,.\n\\end{equation}\n$R_{\\rm growth}$ describes the change in the growth of a small-scale density fluctuation\nat a fixed comoving $k$ in the separate and global universe relative to their own scale\nfactors. $R_{\\rm dilation}$ changes the scale to a fixed wavenumber in the global universe\nor physical wavenumber in each. Finally $R_{\\bar \\rho}$ accounts for the different mean\ndensity of the two universes in the definition of the density fluctuation.\n\n\nTo measure the growth response from SU simulations, we first distribute the dark matter\nparticles onto a $1024^3$ grid by the cloud-in-cell (CIC) density assignment scheme to construct\nthe density fluctuation, and Fourier transform the density fluctuations with FFTW \\cite{fftw}\nto form the power spectrum. For each set of super $\\uparrow$ or sub $\\downarrow$ Jeans\nscale SU simulations with the same initial phases, we estimate the growth response,\n\\begin{equation}\n R_{\\rm growth,\\uparrow\\downarrow} \\equiv \n R_{\\uparrow\\downarrow}\n \\equiv \\frac{d\\ln P_{\\uparrow\\downarrow}}{d\\del{\\uparrow\\downarrow}}\n\\end{equation}\nas\n\\begin{equation}\n \\hat{R}_{\\uparrow\\downarrow}(k,a)=\n \\frac{\\hat{P}_{\\uparrow\\downarrow}(k,a|{\\scriptstyle +}\\del{\\uparrow\\downarrow,0})\n -\\hat{P}_{\\uparrow\\downarrow}(k,a|{\\scriptstyle -}\\del{\\uparrow\\downarrow,0})}\n {2\\hat{P}(k,a) \\del{\\uparrow\\downarrow}(a)} \\,,\n\\end{equation}\nwhere we difference the overdense and underdense pairs for each $|\\del{\\uparrow\\downarrow,0}|$.\nWe then compute the variance of $\\hat{R}_{\\uparrow\\downarrow}$ from the 20 small-box realizations.\n\n\nThe dilation response accounts for the fact that the same comoving $k$ in the SU corresponds\nto a different physical $k$ in the global universe. Given the change in the scale factor from\n\\refeq{aW}, it is analytically related to the local slope in power spectrum as \\cite{Li:2014sga}\n\\begin{equation}\n R_{\\rm dilation}(k,a)=-\\frac{1}{3}\\frac{d\\ln k^3P(k,a)}{d\\ln k} \\,.\n\\label{eq:Rdilation}\n\\end{equation}\nTo compute the dilation response from simulations, we take the log-derivative of the\nmean power spectrum measured from 40 small-box simulations with the global cosmology.\nAs a result, the dilation response is the same in both sub-Jeans and super-Jeans SUs.\nFinally the $\\bar\\rho$ response is due to the change in the definition of a density\nfluctuation\n\\begin{equation}\n \\delta_m \\equiv \\frac{\\delta\\rho_m}{\\bar\\rho_m} = \\delta_{mW} \\frac{ \\bar\\rho_{mW}}{ \\bar\\rho_m} \\,,\n\\end{equation}\nso that to the leading order \\refeq{rhoW} implies\n\\begin{equation}\n R_{\\bar \\rho}(k,a)=2 \\,.\n\\label{eq:Rreference}\n\\end{equation}\nNote that the last two responses, dilation and reference-density, do not involve\nthe SU simulations.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.46\\textwidth]{pk_response_component_z0.pdf}\n\\caption{(Top) Different components of the separate universe power spectrum responses\nat $z=0$: growth response (sub-Jeans, red solid; super-Jeans blue dashed), absolute\nvalue or negative of the dilation response (green dot-dashed), and the reference-density\nresponse (black dotted). (Bottom) Ratios of the mean of the total response to that of\nthe sub-Jeans separate universe. Shaded bands reflect the error on the mean response\nof the simulations. The clear distinction between the sub-Jeans and super-Jeans power\nspectrum responses is the first important result of our separate universe simulations.}\n\\label{fig:resp_tot}\n\\end{figure}\n\n\nThe top panel of \\reffig{resp_tot} shows the various power spectrum responses at $z=0$,\nand the bottom panel shows the ratios of the total power spectrum response, $R_{\\rm tot}$,\nto that of the sub-Jeans SU, $R_{{\\rm tot},\\downarrow}$. We find that the response is\nroughly 2\\% smaller in super-Jeans than in sub-Jeans SUs, and the distinction is statistically\nsignificant. Note also the small errors ($\\sim0.1\\%$ at low-$k$ and $\\sim 0.3\\%$ at high-$k$)\nestimated from small-box SU simulations, demonstrating the power of the SU technique to\nprecisely characterize the response down to arbitrarily small scales.\n\n\nThe fact that the growth response is smaller in super-Jeans than in sub-Jeans SUs can be\nunderstood qualitatively from \\reffig{dm}. Normalized to a given observation redshift,\n$\\delta_m$ in the super-Jeans limit is always smaller in the past than that in the sub-Jeans\nlimit. Consequently, the super-Jeans SU is closer to the global universe along the growth\nhistory, and so the response is smaller.\n\n\nThis difference between super-Jeans and sub-Jeans scales produces an observable change\nin the local power spectrum, and so can in principle be used as a new probe of the sound\nspeed of quintessence. In the real universe, the small-scale power spectrum responds\nto long modes of all scales, the difference of the responses in sub-Jeans and super-Jeans\nlimit would thus appear as the scale-dependent squeezed-limit bispectrum for a fixed\nsmall-scale mode. However in quintessence models with initial curvature perturbations,\nthe predicted amplitude for adiabatic quintessence fluctuations is proportional to\n$(1+w_Q)$ (see Eq.~(95) of Ref. \\cite{Hu:2016ssz}) and so goes to zero as $w_Q \\rightarrow -1$.\nMore generally, this growth history dependence demonstrates that the nonlinear matter power\nspectrum cannot simply be a functional of the linear power spectrum at the same epoch as\nis commonly assumed in simple halo model and nonlinear fitting procedures (see also \\cite{Ma:2006zk}).\n\n\n\\subsection{Perturbation Theory}\n\\label{sec:pt}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnDW.pdf}\n\\caption{The response of the separate universe growth function as a function of the global\nscale factor $a$ in the global universe for super-Jeans (red solid) and sub-Jeans (blue\ndashed) cases. Above the Jeans scale, the response to the same $\\delta_m$ at $a$ is smaller\nthan below.}\n\\label{fig:dlnDW}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnpk_linear.pdf}\n\\caption{Linear perturbation theory predictions for the growth responses compared with\nthe measurements from the 20 small-box separate universe simulations at $z=3$ (top)\nand 0 (bottom). The red solid and blue dashed lines with shaded areas show the sub-Jeans\nand super-Jeans separate universes measurements as in Fig.~\\ref{fig:resp_tot}, whereas\nthe red dot-dashed and blue long-dashed lines show the corresponding linear perturbation\ntheory predictions, i.e.~\\refeqs{Rgrowth}{linear}. Note that the range of $y$-axes is\nsmaller in the top than in the bottom panel. Also the cusp feature at $k\\sim0.078~{\\rm Mpc}^{-1}$\nis a visual artifact due to binning, which we choose to be $\\Delta k= 2\\pi\/L$.}\n\\label{fig:dlnpk_linear}\n\\end{figure}\n\n\nTo better understand the growth responses quantitatively, we compute them in perturbation\ntheory and check their agreement with the SU simulations at various redshifts. In perturbation\ntheory, the effect can be modeled through the SU linear growth function $D_W$ as\n\\begin{equation}\n R_{\\rm growth}(k,a)=\\frac{d\\ln P(k,a)}{d\\ln D_W(a)}\\frac{d\\ln D_W(a)}{d\\delta_m(a)} \\,.\n\\label{eq:Rgrowth}\n\\end{equation}\nIn the linear regime $P(k,a) \\approx P_{\\rm lin}(k,a) \\propto D_W^2(a)$ and so \n\\begin{equation}\n \\frac{d\\ln P(k,a)}{d\\ln D_W(a)}\\approx 2 \\,.\n \\label{eq:linear}\n\\end{equation}\nTo determine the response of $D_W$ in \\refEq{Rgrowth}, we solve \\refEq{DW} with the initial\ncondition \\refEq{D_ic}, and the result is shown in \\reffig{dlnDW}. It approaches the matter\ndominated ($\\Omega_m=1$) limit $13\/21$ at high redshift for both super-Jeans and sub-Jeans\nscale responses, and at low redshift is smaller in the super-Jeans case as expected from\n\\reffig{dm}. In \\reffig{dlnpk_linear}, we compare the measured power spectrum response in\nsub-Jeans and super-Jeans SUs to the corresponding linear perturbation theory predictions,\ni.e.~\\refeqs{Rgrowth}{linear}. We find that in both cases the linear perturbation theory\nagrees with the measured responses in the linear regime, i.e., at sufficiently low $k$ or\nhigh $z$.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnpk_1loop.pdf}\n\\caption{Same as \\reffig{dlnpk_linear}, but for the 1-loop predictions,\ni.e. \\refeq{Rgrowth} and \\refeq{1loop}.}\n\\label{fig:dlnpk_1loop}\n\\end{figure}\n\n\nHowever, as we move to lower redshift as well as higher $k$, the measured responses\nbecome nonlinear and perturbation theory predictions deviate from SU simulation\nmeasurements. Unlike perturbation theory, the SU response calibration is not limited\nto large scales. On the other hand, we can understand the onset of nonlinearity in\nthe simulations through higher order perturbation theory. The 1-loop power spectrum\nfrom standard perturbation theory is given by (see e.g. Ref.~\\cite{Jeong:2006xd})\n\\begin{equation}\n P_{\\rm 1-loop}(k,a)=P_{\\rm lin}(k,a)+P_{22}(k,a)+2P_{13}(k,a) \\,,\n\\end{equation}\nwhere the nonlinear corrections $P_{22}$ and $P_{13}$ are proportional to $D_W^4$\nif $\\Omega_{mW}(a_W)\/f_{W}^2(a_W)\\approx1$.\nTherefore,\n\\begin{equation}\n \\frac{d\\ln P_{\\rm 1-loop}(k,a)}{d\\ln D_W(a)}\n =2\\left[1+\\frac{P_{22}(k,a)+2P_{13}(k,a)}{P_{\\rm 1-loop}(k,a)}\\right] \\,,\n \\label{eq:1loop}\n\\end{equation}\nwhich now is a function of $k$. Note that in the global cosmology $\\Omega_m\/f^2=1.034$\nat $z=3$ and 1.203 at $z=0$, and the standard perturbation theory should work better\nat $z=3$ than at $z=0$. Since the long-wavelength perturbation we consider is small\n($|\\delta_{\\uparrow\\downarrow0}|=0.01$), the standard perturbation theory should work\nas well in both the sub-Jeans and super-Jeans SUs as the global universe.\n\nIn \\reffig{dlnpk_1loop} we plot the 1-loop predictions in sub-Jeans (red dot-dashed)\nand super-Jeans (blue long-dashed). We find that the 1-loop predictions extend the\nagreement with the $N$-body measurement to smaller scales compared to the linear\npredictions. More precisely, the difference between the 1-loop model and the measurement\nat $z=3$ ($z=0$) is 1\\% (3\\%) at $k\\sim0.1~{\\rm Mpc}^{-1}$ and 4\\% (6\\%) at $k\\sim0.2~{\\rm Mpc}^{-1}$.\nAt even smaller scale or lower redshift, the nonlinearity is too large to be modeled\nby the 1-loop perturbation theory. The SU simulation calibration technique itself is\nnot limited in wavenumber and our $N$-body implementation is instead only limited by\nresolution as well as the lack of baryonic and astrophysical modeling in the deeply\nnonlinear regime.\n\n\n\\subsection{Position-Dependent Power Spectrum}\n\\label{sec:ibn}\nThe power spectrum response can also be tested in simulations and observed in surveys \nthrough the position-dependent power spectrum. As a simulation based test, it also serves\nto check the SU calibration of the power spectrum response deep into the nonlinear regime.\n\n\nSpecifically we compare the response measured from the small-box SU simulations to the\nsqueezed-limit position-dependent power spectrum measured from the big-box simulations\nwith the global cosmology (without a uniform long-wavelength density fluctuation). In\nthe latter, we assume that the dark energy does not cluster with matter and so its\nposition-dependent power spectrum should match the sub-Jeans SU prediction.  \n\n\nThe procedure of measuring the position-dependent power spectrum is explained in detail\nin \\cite{Chiang:2014oga}. In short, we first distribute the dark matter particles onto\na $2048^3$ grid by the CIC density assignment scheme to construct the density fluctuation.\nWe next divide the big-box simulations in each dimension by 8, so there are  $N_s=512$\nsubvolumes in total  with  comoving side length of $L=350$ Mpc. In each subvolume centered at\n${\\mathbf r}_L$, we measure the local power spectrum as $\\hat{P}(k,a|{\\mathbf r}_L)$\nand the mean overdensity (with respect to the entire box) as $\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)$\nand construct \n\\begin{equation}\n \\frac{\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{P}(k,a|{\\mathbf r}_L)\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)}\n {\\left[\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{P}(k,a|{\\mathbf r}_L)\\right]\n \\left[\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{\\bar{\\delta}}_m^2({\\mathbf r}_L)\\right]} \\,,\n\\end{equation}\nwhere the summation is over the 512 subvolumes in one big-box realization. The correlation\nbetween $\\hat{P}(k,a|{\\mathbf r}_L)$ and $\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)$ quantifies\nthe integrated bispectrum, and in the squeezed limit where $k\\ll1\/L$ the integrated bispectrum\ncan be understood as the {\\it total} response of the power spectrum to the long-wavelength\noverdensity.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{ibn_z0.pdf}\n\\caption{Comparison of the squeezed-limit position-dependent power spectrum of big-box,\nglobal simulations (green dot-dashed), and the total power spectrum response of sub-Jeans\n(red solid) as well as super-Jeans (blue dashed) separate universe simulations at $z=0$.\nThe shaded areas show the error on the mean. This figure summarizes our main power spectrum\nresponse results: the position-dependent power spectrum and the sub-Jeans power spectrum\nresponse agree significantly better than the difference in response across the Jeans scale\nconfirming the scale dependence in this observable deep into the nonlinear regime.}\n\\label{fig:ibn}\n\\end{figure}\n\n\\refFig{ibn} shows the comparison at $z=0$ between the total power spectrum response from\nthe previous section (red solid for sub-Jeans and blue dashed for super-Jeans cases) and\nthe position-dependent power spectrum  (green dot-dashed) averaged over the 20 realizations\nwith its error. To reach the squeezed limit, we require $k\\gtrsim100\/L\\sim0.3~{\\rm Mpc}^{-1}$,\nand in this regime the agreement with the SU response is better than a percent. This agreement\nis significantly better than the difference between the super-Jeans and sub-Jeans power spectrum\nresponses and thus verifies the SU calibration technique. With the SU technique tested into\nthe nonlinear regime, we can apply these results to the super-Jeans case of the position-dependent\npower spectrum without the need for costly simulations that include dark energy clustering.\n\n\n\\section{Scale-Dependent Halo Bias}\n\\label{sec:bias}\nThe SU simulations also calibrate the response of the halo mass function to a long-wavelength \nmode and hence the bias of the halo number density due to that mode. In \\refsec{responsebias},\nwe review the technique for measuring halo bias from SU simulations and show that in the\nquintessence model it acquires a scale dependence at the Jeans scale. In \\refsec{clusteringbias},\nwe test this response bias in the SU simulations against the clustering bias extracted from\n40 small-box global simulations. We discuss the implications of scale dependence for the\ntemporal nonlocality of halo bias and the observability of features in the halo power\nspectrum in \\refsec{interpretation}.\n\n\n\\subsection{Response Bias}\n\\label{sec:responsebias}\n\nThe linear density bias $b_1(M)$ of halos of mass $M$ can be defined as the response\nof the differential halo abundance $n_{\\ln M}=dn\/d\\ln M$ to the long-wavelength mode\n\\begin{equation}\n b_1(M) \\equiv \\frac{d\\delta_h}{d\\delta_m} = \\frac{d\\ln n_\\lnM}{d\\delta_m} \\,,\n \\label{eq:biasasresponse}\n\\end{equation}\nwhich we call ``response bias''. Thus by measuring the response of the halo mass\nfunction in the SU simulations we have a direct calibration of response bias\n\\cite{Li:2015jsz,Lazeyras:2015lgp,Baldauf:2015vio}. Note that the derivative in\n\\refeq{biasasresponse} is evaluated at a fixed time, but will depend on the whole\ngrowth history of $\\delta_m(a)$. This temporal nonlocality implies that response\nbias can be scale dependent if that growth history is also scale dependent. For\nquintessence, the SU simulations allow us to calibrate the bias above and below\nthe Jeans length of quintessence without direct simulations of its clustering properties. \n\n\nAs discussed in Ref.~\\cite{Li:2015jsz}, response bias largely reflects the change\nin the masses of halos due to the same local change in growth that affects the power\nspectrum. The enhanced growth in $\\delta_m>0$ regions makes halos more massive locally\nthan in their $\\delta_m<0$ counterparts. Hence halos of a fixed mass are associated\nwith the more abundant lower peaks in the initial density field in the former and\nthe less abundant higher peaks in the latter. This also means that measuring the\nchange in abundance of halos in fixed mass bins between the SU simulation pairs is\nan inefficient way to quantify response bias. Halos with small changes in mass across\nwide mass bins would register their response only when individual halos move across\nmass bins.\n\n\nWe instead adopt abundance matching as introduced in Ref.~\\cite{Li:2015jsz}, which\nwe now summarize. By finding the mass threshold above which the cumulative abundance\nis fixed, we largely eliminate the sampling noise from the discrete nature of halos.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{s_z0.pdf}\n\\caption{Threshold mass shift as a response of varying $\\delta_m$ at fixed cumulative abundance\nat $z=0$. The solid line and shaded region show the smoothed estimate and the bootstrap error.}\n\\label{fig:s_z0}\n\\end{figure}\n\n\nSpecifically for either the sub-Jeans or the super-Jeans case, we first combine halo catalogs\nof all realizations of the same $\\delta_m$ in the small-box suite. The masses of the $i^{\\rm th}$\nmost massive halo $M_i^\\pm$ from the $\\delta_m = \\pm 0.01$ SU simulations determine the\ndiscrete threshold mass shift\n\\begin{equation}\n s_i(\\lnM_i) = \\frac{\\lnM_i^+ - \\lnM_i^-}{2|\\delta_m|} \\,,\n\\end{equation}\nwhere $M_i$ is the geometric mean of $M_i^+$ and $M_i^-$. We then use the smoothing spline\ntechnique to estimate the ensemble average threshold mass shift $\\hat s(\\ln M)$ as well as\nthe cumulative halo abundance above threshold mass $\\hat n(\\ln M)$. \\refFig{s_z0} shows the\nmass shift measured  from 20 sub-Jeans and super-Jeans SU simulations at $z=0$ as a function\nof halo mass. We find that the mass shift due to varying $\\delta_m$ is smaller above the\nJeans scale, which reflects the fact that its growth history makes it closer to the global\nuniverse than below the Jeans scale.\n\n\nThe halo mass function follows as the derivative of the cumulative mass function\n$\\hat n_\\lnM=-d\\hat n\/d\\lnM$. We can then estimate the Lagrangian halo bias above\nthreshold mass $M$ as\n\\begin{equation}\n \\hat{\\bar{b}}_1^\\Lr(M) = \\frac{\\hat n_\\lnM(\\ln M)\\,\\hat s(\\ln M)}{\\hat n(\\ln M)} \\,.\n\\label{eq:am}\n\\end{equation}\nThis quantity is the Lagrangian bias since the SU simulations are performed with the\nsame comoving rather than physical volume. The dilation of the volume from the change\nin scale factors brings the cumulative Eulerian bias to\n\\begin{equation}\n \\hat{\\bar{b}}_1(M) =1+ \\hat{\\bar{b}}_1^\\Lr(M)\\,.\n \\label{eq:eulerianb}\n\\end{equation}\nIn \\reffig{b_resp} we compare the response bias on super-Jeans (blue dashed) and sub-Jeans \n(red solid) scales as a function of halo mass at $z=0$. The bias at a fixed mass is smaller\nin the super-Jeans case. Just like for the power spectrum response, above the Jeans scale\nfor the same final $\\delta_m$, the SU is closer to global at high redshift. Thus the change\nin growth and the consequent change in halo masses and abundances is smaller. We find that\nthe mild mass dependence of the fractional difference between the super-Jeans and sub-Jeans\nresponse biases is due mainly to the dilation effect in \\refeq{eulerianb}, since that of\nthe Lagrangian bias is fairly mass independent due to the similar shapes of the mass shift\ndisplayed in \\reffig{s_z0} (see also \\reffig{bLversusmodels}).\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.46\\textwidth]{b_response_z0.pdf}\n\\caption{(Top) The $z=0$ response biases measured from 20 sub-Jeans (red solid) and\nsuper-Jeans (blue dashed) separate universe simulations. The lines and shaded areas\nshow the smoothed estimate and the bootstrap error. (Bottom) The ratios of the response\nbiases to that of the sub-Jeans response bias. The difference between the sub-Jeans\nand super-Jeans response biases, which indicates that the linear halo bias is scale\ndependent in the presence of the scale-dependent growth, is the second central result\nof our separate universe simulations.}\n\\label{fig:b_resp}\n\\end{figure}\n\n\n\\subsection{Clustering Bias}\n\\label{sec:clusteringbias}\n\nTo verify the SU calibration of halo bias through the mass function response, we can\ncompare it to how linear halo bias is commonly measured from the two-point statistics,\nwhich we call clustering bias\n\\begin{equation}\n\\bar b_1(M) = \\lim_{k\\to0}\\frac{P_{hm}(k;M)}{P_{mm}(k)} \\,,\n \\label{eq:biasascrosspower}\n\\end{equation}\nwhere $P_{hm}$ is the cumulative halo number density cross power spectrum with the\nmatter density. Where no confusion should arise, we omit the $M$ argument of the\ncumulative bias. Above the Jeans scale of quintessence, this approach would require\nsimulations of quintessence clustering even for linear halo bias. Below the Jeans\nscale, we can test the equivalence of response and clustering bias with global\nsimulations where quintessence enters only at the background level.  \n\n\nIn order to extract the $k\\rightarrow 0$ limit, we first compute\n\\begin{equation}\n \\bar{q}(k)=\\frac{{P}_{hm}(k)}{P_{mm}(k)} \\,,\n\\end{equation}\nfor each of the 40 simulations of the global cosmology for a set of mass thresholds.    \nMotivated by Ref.~\\cite{Assassi:2014fva}, we fit $\\bar{q}(k)$ to the model\n\\begin{equation}\n \\bar{b}(k)=\\bar{b}_1+\\sum_{i=1}^n \\bar{b}_{k^{2n}}k^{2n} \\,,\n\\end{equation}\nwhere we treat $\\bar{b}_{k^{2n}}$ as nuisance parameters that absorb the loop\ncorrections in the large-scale limit. We then get the best-fit bias parameters\nby minimizing\n\\begin{equation}\n \\chi^2=\\sum_{k}^{k_{\\rm max}}\\frac{[\\bar{q}(k)-\\bar{b}(k)]^2}{\\sigma^2[\\bar{q}(k)]} \\,,\n\\label{eq:chi2}\n\\end{equation}\nwhere $\\sigma^2[\\bar{q}(k)]$ is the variance of $\\bar{q}(k)$ measured from 40 global\nsmall-box simulations.\n\n\nTo ensure the robustness of the fitted clustering bias, especially as compared with the\nsmall predicted difference between sub-Jeans and super-Jeans response biases, we examine\nthe bias models with $n=0$, 1, and 2 for various $k_{\\rm max}$. We seek consistent result for\ndifferent bias models (different $n$) and $k_{\\rm max}$. The general principle is that the\nlarger the $k_{\\rm max}$, the larger the $n$ required to account for the nonlinearity and to\navoid underfitting. Conversely, for models with $n>0$ $k_{\\rm max}$ cannot be too small or the\nfit would suffer from overfitting.\n\n\nWith each bias model and  $k_{\\rm max}$, we visually inspect its goodness of fit to $\\bar{q}(k)$\nfor various threshold halo masses. We find that across two decades in halo mass\n($2\\times10^{13}-2\\times10^{15}\\,M_\\odot$), the bias models of $n=0$, 1, and 2 with the\nbiases fitted to $k_{\\rm max}=0.014-0.028\\,{\\rm Mpc}^{-1}$, $0.042-0.049\\,{\\rm Mpc}^{-1}$,\nand $0.056-0.07\\,{\\rm Mpc}^{-1}$ are in agreement with the mean $\\bar{q}(k)$, and the\nagreement even extends to $k>k_{\\rm max}$. This shows that the fit is free from overfitting\nand underfitting problems. For a given halo mass, the best-fit clustering bias varies\nup to 0.2\\%, 0.5\\%, and 2\\% among different $n$ and $k_{\\rm max}$ at $2\\times10^{13}\\,M_\\odot$,\n$2\\times10^{14}\\,M_\\odot$, and $2\\times10^{15}\\,M_\\odot$, respectively. Given the fact\nthat the clustering bias is stable for various bias models and fitting range, we conclude\nthat systematic error due to $n$ and $k_{\\rm max}$ is at most comparable to our statistical\nerror, and is the largest at the high-mass end at which the statistical error is also large.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{b_clustering_ratioonly_z0.pdf}\n\\caption{The ratios of the linear biases to that of the sub-Jeans response bias at $z=0$.\nThe red solid and blue dashed lines show the sub-Jeans and super-Jeans response biases\nmeasured from 20 separate universe simulations, whereas the green dot-dashed line shows\nthe clustering bias measured from 40 global simulations. The error of the clustering bias\nis measured from the scatter of the 40 simulations. This figure summarizes the main results\non halo bias: the agreement between the clustering bias measured from global simulations\nand the sub-Jeans response bias verifies the observable difference in halo bias across the\nJeans scale inferred from the separate universe simulations.}\n\\label{fig:b_clus}\n\\end{figure}\n\n\nIn \\reffig{b_clus} we shows our fiducial results of the clustering bias measurement for\nthe quadratic model ($n=1$) with $k_{\\rm max}=0.49\\,{\\rm Mpc}^{-1}$, which gives the smallest\nstatistical errors. We find that the clustering bias is in good agreement with the sub-Jeans\nresponse bias across two decades in halo mass, confirming the validity of the SU technique.\nThis agreement is substantially better than the difference between the super-Jeans and sub-Jeans\nresponse bias at low- and mid-mass regime, even after including the systematic differences\nbetween the fitting techniques.\n\n\nTo further test robustness of the scale-dependent bias result, we also try the halo\nfinding algorithm provided in Ref.~\\cite{Li:2015jsz}, another spherical overdensity finder\nsimilar to that in Ref.~\\cite{Tinker:2008ff}. We find that the clustering bias is statistically\nin equally good agreement with the sub-Jeans response as well.\n\n\nWith this verification of the SU calibration of halo bias, our results represent the first\nsimulation confirmation of scale-dependent halo bias from scale-dependent growth. A related\neffect on the void bias has been measured in the simulations with cold dark matter and massive\nneutrinos \\cite{Banerjee:2016zaa}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{phh_sup_to_sub.pdf}\n\\caption{Fractional difference or ``step'' across the Jeans scale in the matter (red solid,\n\\refeq{Smm}) versus the halo (blue dashed, \\refeq{Shh}) power spectra. The step in the halo\npower spectrum is reduced by approximately a factor of 2 compared with the matter power\nspectrum at high mass where the Lagrangian bias contribution dominates. Bias prescriptions\nwhich assume locality at either the observed or initial redshift predict the same step or\nzero step respectively at high masses.}\n\\label{fig:phh_ratio}\n\\end{figure}\n\n\\subsection{Scale-Dependent Bias and Power Spectra}\n\\label{sec:interpretation}\n\n\nSince the halo bias is smaller above versus below the Jeans scale of quintessence, its\nscale dependence counters the growth rate effects in the matter power spectrum. This\nis especially true at high masses where the Lagrangian bias dominates. The change in\nthe linear growth function above $(D^\\uparrow)$ versus below $(D^\\downarrow)$ the Jeans\nscale leads to a step in the linear matter power spectrum of approximately\n\\begin{equation}\n\\label{eq:Smm}\nS_{mm} \\equiv \n2 \\frac{D^\\uparrow - D^\\downarrow}{D^\\downarrow}\\,,\n\\end{equation}\nwhereas the step in the cumulative halo power spectrum is\n\\begin{equation}\nS_{hh}\n \\equiv\n  2 \\frac{D^\\uparrow \\bar b_1^\\uparrow - D^\\downarrow \\bar b_1^\\downarrow}{D^\\downarrow \\bar b_1^\\downarrow} \\,.\n  \\label{eq:Shh}\n\\end{equation}\nIn \\reffig{phh_ratio}, we show the amplitude of these steps as a function of mass.\nAt the high mass end the halo power spectrum has half the step amplitude of the\nmatter power spectrum.\n\n\nThis result not only confirms that scale-dependent linear growth leads to scale-dependent\nbias, but it does so in a way that both reduces the observability of features in the halo\npower spectrum \\cite{LoVerde:2014pxa} and violates principles that underlie simple models\nfor bias. It is commonly assumed that the statistics of halos at any observation epoch is\ndetermined solely by the statistics of the linear density field at a single epoch, and\nhence bias is scale-free with respect to the matter power spectrum at that epoch. For\nmodels with scale-dependent growth, this epoch is commonly taken to be  the initial epoch\nfor models where the growth becomes scale-free during matter domination \\cite{Parfrey:2010uy}.\n\n\nFor example in the excursion set, Lagrangian bias is given by the conditional probability\nthat the initial density field crosses some barrier at a smoothing scale $R_S$ corresponding\nto the mass $M$ at the background density given that it takes the value $\\delta_m$ at some\nlarger scale $R_L$ via a random walk between the two. For our quintessence case where these\nscales are arbitrarily well separated by our SU assumption, the lack of correlation in the\nGaussian random initial conditions between these scales means that halo bias is local in the\ninitial density field. Specifically, the conditional probability cannot depend on steps in\nthe random walk with $R>R_L$, and hence whether $\\delta_m$ was achieved from super-Jeans or\nsub-Jeans scale fluctuations. This holds regardless of the shape of the barrier, its dependence\non the redshift of observation, or some putative intermediate epoch of halo formation. As\na result, Lagrangian halo bias should be scale-free with respect to the matter power spectrum\nat the initial epoch.\n\n\nAs emphasized in Ref.~\\cite{Parfrey:2010uy}, even if the Lagrangian bias is scale-free with\nrespect to the initial power spectrum, it becomes scale dependent with respect to the matter\npower spectrum at the observation epoch. This effect is solely due to the scale-dependent\ngrowth in the latter, and hence the scale dependence takes a simple and specific form\n$b_1^{L\\uparrow} = (D^\\downarrow\/D^\\uparrow) b_1^{L\\downarrow}$. In \\refapp{bias_model},\nwe review this construction in more detail. In the quintessence model this means that the\nstep in the halo power spectrum should be absent when the Lagrangian bias dominates\n$\\lim_{M\\rightarrow \\infty}S_{hh}=0$ (see \\refeq{Shh}). Our results significantly violate\nthis prediction.\n\n\nSimilarly models of halo bias that rely on a universal mass function ansatz, characterize\nthe bias as its derivative with respect to $\\delta_m$ at the observation epoch. If this\nderivative depends {\\it only} on the local density field $\\delta_m$ at the observation epoch,\nfor example by assuming a change in the spherical collapse threshold $d\\delta_c\/d\\delta_m=-1$\n(see \\refapp{bias_model}), then the Lagrangian bias would be local and hence scale-free with\nrespect to the matter power spectrum at the observation epoch. Our results for scale-dependent\nbias directly violate this prediction and are essentially half-way between these two extreme\nmodels. We find that halo bias is nonlocal in time and cannot be characterized by the statistics\nof the density field at a single epoch, initial or observed when there is scale-dependent\nlinear growth.\n\n\nIn \\refapp{bias_model}, we show that encapsulating the dependence on the growth history of\n$\\delta_m(a)$ through its impact on the spherical collapse threshold at the observation epoch\nand assuming a universal mass function characterizes the quintessence SU simulation results\nbetter than either of these simplistic models. However given the assumptions underlying this\ntype of modeling, its validity in other contexts should be tested directly in simulations.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\nQuintessence dark energy provides an arena to explore the response of small-scale\nobservables to the amplitude, scale, and growth history of long-wavelength fluctuations\nwith the separate universe technique. In the presence of quintessence fluctuations,\nthe growth of long-wavelength fluctuations differ above and below its Jeans scale.   \nWe verify that even below the Jeans scale, where a naive separate universe picture\ndoes not strictly apply because the local curvature evolves due to non-gravitational\nforces which keep the quintessence smooth, the response of small-scale observables\ncan still be accurately modeled by a modified expansion history alone. One implication\nof this finding is that halo bias is not directly a response of halo number density\nto the local curvature, but rather to the local expansion history.\n\n\nUsing this technique, we show that in the presence of the scale-dependent growth,\nthe local power spectrum and halo mass function acquire a dependence on the scale\nof the long-wavelength mode. Equivalently, the squeezed bispectrum and halo bias\nbecome scale dependent. To our knowledge, our results are the first verification\nof scale-dependent bias from scale-dependent growth using simulations. Moreover\nthey violate predictions of models where bias is effectively local in the density\nfield at a single epoch, initial or observed, and show that halo bias is temporally\nnonlocal. Likewise the nonlinear matter power spectrum cannot simply be a function\nof the linear power spectrum at the same epoch.\n\n\nSpecifically, we use the separate universe (SU) technique to perform $N$-body\nsimulations in the sub-Jeans and super-Jeans SUs. By differencing pairs of\noverdense and underdense SU simulations with the same Gaussian realizations\nof initial phases, much of the sample variance is canceled, and so we can\nprecisely characterize the responses of the power spectrum (which is equivalent\nto the squeezed-limit bispectrum) and the halo mass function (which gives the\nlinear halo bias) to the long-wavelength matter fluctuation.\n\n\nWe validate the SU approach by comparing to perturbation theory predictions\nfor the power spectrum response in both the super-Jeans and sub-Jeans limits\n(see \\reffigs{dlnpk_linear}{dlnpk_1loop}). Since it is the sub-Jeans limit\nwhere the SU technique might naively fail, we further test it with direct\nsimulations that possess long-wavelength matter modes in big-box simulations\nwith smooth dark energy. We find that the squeezed-limit position-dependent\npower spectrum measured from the big-box simulations agrees with the power\nspectrum response to the resolution limit $k\\sim1\\,{\\rm Mpc}^{-1}$. Similarly,\nthe clustering bias is statistically consistent with the response bias across\ntwo decades in halo mass ($\\sim\\!10^{13}\\!-\\!10^{15}\\,M_\\odot$). Thus, with\nthe SU technique verified into the nonlinear regime, we can robustly assess\nthe scale-dependence of the power spectrum and halo density responses across\nthe Jeans scale without costly simulations that include quintessence clustering.\n\n\nWe show that for both responses there is a statistically significant distinction\nbetween sub-Jeans and super-Jeans SUs at $z=0$. More precisely, the power spectrum\nresponse in the super-Jeans SU is roughly 2\\% smaller than that in the sub-Jeans\nSU for $k\\lesssim1\\,{\\rm Mpc}^{-1}$; the halo bias in the super-Jeans SU is roughly\n1\\% and 3\\% smaller than that in the sub-Jeans SU for halo mass of $2 \\times 10^{13}$\nand $2\\times 10^{15}~M_\\odot$ respectively. The fact that the response is smaller\nin the super-Jeans SU is because quintessence enhances the growth of matter\nfluctuations there, and so the super-Jeans overdensity was smaller in the past.\nThese key SU results, along with the comparison to the global simulations,\nare summarized in \\reffig{ibn} and \\reffig{b_clus}.\n\n\nMore generally, this dependence on the growth history of the long wavelength fluctuation\nindicates that the response of small scale observables is nonlocal in time. In particular,\nthe statistically significant difference between sub-Jeans and super-Jeans response biases\nmeasured in our SU simulations falsifies the standard Lagrangian picture where the statistics\nof halos at any observation epoch is determined solely by the statistics of the linear density\nfield at a single epoch.\n\n\nThese effects are in principle important for interpreting observational tests of quintessence\nclustering from galaxy surveys and their cross correlation with the CMB (e.g. \\cite{Bean:2003fb,Hu:2004yd}).\nIn particular, the step feature in the halo power spectrum is smaller by up to a factor of 2\ncompared with the matter. However these corrections, while significant relative to the clustering\neffects on the matter power spectrum itself, are small in an absolute sense for observationally\nviable dark energy equations of state (i.e. $w_Q\\approx-1$) in the absence of quintessence\nisocurvature fluctuations \\cite{Gordon:2004ez,Hu:2016ssz}. \n\n\nOn the other hand, the same technique which has been validated here using quintessence,\ncan be applied to more observationally viable cosmological models, such as those with\nmassive neutrinos. Massive neutrinos cluster with dark matter on large scales, but their\nfree streaming sets an effective Jeans scale. This would generate not only a feature in\nthe two-point function of the total matter \\cite{Lesgourgues:2006nd}, but also influence\nthe high-order statistics \\cite{Shoji:2009gg,Blas:2014hya,Fuhrer:2014zka,Levi:2016tlf}\nas well as the halo bias \\cite{LoVerde:2014pxa}. We intend to apply the SU technique\nto study how massive neutrinos affect the small-scale structure formation in a\nfuture work.\n\n\n\\acknowledgements\nWe thank Eiichiro Komatsu and Fabian Schmidt for useful discussions.\nWe would also like to thank Alexander Knebe for guiding us to implement the dark energy model into Amiga Halo Finder. \nWH thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, \nwhere part of this work was completed.\nResults in this paper were obtained using the high-performance computing system\nat the Institute for Advanced Computational Science at Stony Brook University\nand with the computation and storage resources\nprovided by the University of Chicago Research Computing Center.\nCC and ML are supported by grant NSF PHY-1316617.\nWH was supported by U.S.~Dept.\\ of Energy contract DE-FG02-13ER41958,\nNASA ATP NNX15AK22G, and  the Kavli Institute for Cosmological Physics\nat the University of Chicago through grants NSF PHY-0114422\nand NSF PHY-0551142.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\newlength{\\szerkol}\n\n\\setlength{\\szerkol}{0.5\\textwidth}\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1.em}\\includegraphics[width=1\\szerkol,clip,viewport=2 150 575 690]{t1png_withsn.ps} &\n\\hspace{-1.5em}\\includegraphics[width=1\\szerkol,clip,viewport=2 150 575 690]{t2png_withsn.ps}\\\\\n\n\\end{tabular}\n\\end{center}\n\\caption{\nLow resolution  {\\sc Hi} map\n(the beam size of $72\\arcsec\\times62\\arcsec$)\nsuperimposed on an optical image of M74.\nThe position of {SN\\,2002ap} is marked as the red dot\n(credit: Kamphuis \\& Briggs, 1992, reproduced with permission $\\copyright$ ESO).\nLeft: Zeroth moment map (integrated emission).\nRight:\nFirst moment map (velocity fields).\nThe asymmetric tail with an irregular velocity field is visible at the south-western outskirt of the atomic disc (around the position of $\\mbox{R.A.}=1^h 33^m20^s$, $\\mbox{Dec.}=15^\\circ 25^m$).\nNorth is up and east is to the left.\n}\n\\label{fig:lowres}\n\\end{figure*}\n\n\n\\setlength{\\szerkol}{0.3\\textwidth}\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom1.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom2.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom1.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom2.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_CO21_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_CO21_mom1.eps} & \n\\end{tabular}\n\\end{center}\n\\caption{Gas distribution in M74. \nTop and middle: \n{\\sc Hi} data with a resolution of \n$11.9\\arcsec\\times9.3\\arcsec$ and\n$6.9\\arcsec\\times5.6\\arcsec$, respectively \\citep{walter08}. Bottom: CO(2-1) data with a resolution of $13.4\\arcsec$ \\citep{leroy09}. \nLeft: Zeroth moment maps (integrated emission).\nMiddle:\nFirst moment maps (velocity fields) relative to $z=0.00219$ (656.545\\,{\\mbox{km\\,s$^{-1}$}}).\nRight: Second moment maps (velocity dispersion). The positions of SNe are marked by grey circles. The lines outline the main spiral arm. \nEach panel is 10{\\arcmin} per side. North is up and east is to the left.\n}\n\\label{fig:image}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=\\szerkol,clip]{M74_UVW2_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_Ha_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_36_cont.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_80_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_160_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_31_cont.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Ultraviolet, H$\\alpha$, mid-IR, far-IR, and radio images of M74.\nThe positions of SNe are marked by grey circles. The lines outline the main spiral arm.\nEach panel is 10{\\arcmin} per side. North is up and east is to the left.\n}\n\\label{fig:image2}\n\\end{figure*}\n\nThe nature of various types of supernovae (SNe) carry crucial information about stellar evolution.\nA subclass of SNe with no hydrogen, helium, or silicon lines in the spectrum (known as type Ic) are believed to be explosions of stars born with very high masses.\nThose exhibiting broad emission lines, indicating high velocities of the ejected material (up to a few $10^4\\,\\mbox{km\\,s$^{-1}$}$), are called `hypernovae' or SNe type Ic-BL (broad lined). Some of these SNe also show relativistic ejecta, for example SN\\,1998bw \\citep[GRB\\,980425][]{galamanature} and 2009bb \\citep{soderberg10}. These relativistic features are interpreted as a jet, and indeed some Ic-BL SNe have been associated with gamma-ray bursts (GRBs; \\citealt{hjorthsn}).\n\nObservations of atomic and molecular gas (through 21\\,cm {\\sc Hi} and carbon monoxide [CO] lines, respectively)  in  host galaxies of GRBs and SNe have recently been used to learn about the nature of the explosions themselves, as well as the star formation event during which their progenitors were born.\n\\citet{michalowski15hi,michalowski16,michalowski18} and \\citet{arabsalmani15b,arabsalmani19} showed that GRBs and a relativistic SN type Ic-BL exploded close to the most {\\sc Hi}-rich region of their hosts, which was interpreted as being the result of a recent gas accretion or a galaxy merger.\nWhile this conclusion was based on very small samples, if substantiated it would have important consequences for our understanding of the conditions necessary for such explosions, as well as for triggering star formation in general. This motivates us to study the gas properties of another Ic-BL SN (to date, amongst the hosts of Ic-BL SNe, atomic gas was studied in only one case;\n\\citealt{michalowski18}).\n\n{SN\\,2002ap}  \\citep{nakano12circ} \nexploded $\\sim4.7\\arcmin$ ($\\sim12.7$\\,kpc)\nfrom the centre of \\object{Messier 74} (\\object{M74} or \\object{NGC\\,628}) \nand was classified as a type Ic-BL \\citep{mazzali02,kinugasa02,\ngalyam02,foley03%\n}. Its estimated progenitor mass is $20$--$25\\,\\mbox{$M_\\odot$}$, lower than other hypernovae, including SN\\,1998bw \\citep{mazzali02}. {SN\\,2002ap} was also shown to have only modest relativistic ejecta, and hence no detectable jet \\citep{berger02}. The progenitor was proposed to be a Wolf-Rayet (WR) star or a massive star in an interacting binary \\citep{smartt02,wang03%\n}.\n\nM74 has hosted three other known SNe: 2003gd (type IIP; $\\sim2.7\\arcmin$ or $\\sim7.3$\\,kpc from the galaxy centre; \\citealt{hendry05}), 2013ej (IIP; $\\sim2.2\\arcmin$ or $\\sim5.9$\\,kpc;  \\citealt{valenti14}), and 2019krl (IIn; $1.9\\arcmin$ or $\\sim5.2$\\,kpc; \\citealt{ho19rep,\nandrews19atel}). The progenitor of SN\\,2003gd was confirmed to be an M-type supergiant with a mass of $\\sim8\\,\\mbox{$M_\\odot$}$,\nby examining the SN position in pre- and post-explosion images, which revealed that this star was missing in the latter\n\\citep{maund09,vandyk03,smartt04}. In a similar way, the mass of the progenitor of SN\\,2013ej was estimated to be $8.0$--$15.5\\,\\mbox{$M_\\odot$}$\\citep{fraser14,mauerhan17}.\n \nThe objectives of this paper are to {\\it i}) test whether {SN\\,2002ap} and other SNe in M74 were born in concentrations of gas indicating a recent gas accretion, and {\\it ii}) investigate what this tells us about the formation of SN progenitors.\nWe adopt a redshift of M74 of $z=0.00219$ \\citep{lu93}, a distance of 9.4\\,Mpc, and a corresponding scale of 2.7 kpc arcmin$^{-1}$.\nThis  assumes a cosmological model with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$,  $\\Omega_\\Lambda=0.7$, and $\\Omega_{\\rm m}=0.3$.\n\n\n\n\n\n\\section{Selection and data}\n\\label{sec:data}\n\n\n\n{The supernova SN\\,2002ap} and its host galaxy M74 were selected as part of a larger study of gas in SN hosts (Gotkiewicz \\& Micha\\l owski, in prep.). We investigated all known SNe up to Aug 2018 with redshifts $z<0.1$ from the Open Supernova Catalog\\footnote{{\\tt https:\/\/sne.space}} \\citep{snespace} and searched the NASA\/IPAC Extragalactic Database (NED) for {\\sc Hi} data for their hosts.  {SN\\,2002ap} was the only SN type Ic-BL that satisfied these criteria.\n\nThe archival {\\sc Hi} and CO data for M74 are shown in Figs.~\\ref{fig:lowres} and \\ref{fig:image}, while the rest of the data are shown in Fig.~\\ref{fig:image2}.\n{\\sc Hi} data are from \\citet[][a resolution of $72\\arcsec\\times62\\arcsec$]{kamphuis92}  and The {\\sc Hi} Nearby Galaxy Survey \\citep[THINGS;][a resolution of $11.9\\arcsec\\times9.3\\arcsec$ and\n$6.9\\arcsec\\times5.6\\arcsec$]\n{walter08}. The CO(2-1) data are from the HERA CO Line Extragalactic Survey \\citep[HERACLES;][a resolution of $13.4\\arcsec$]{leroy09}. The H$\\alpha$ image is from \\citet{marcum01}. In addition, we use the following continuum data: \nthe {\\it Swift} \\citep{swift,uvot} \nUVW2 0.2\\,{\\mbox{$\\mu$m}} image \\citep{brown14};\n{\\it Spitzer} \\citep{spitzer,irac} \n3.6 and 8.0\\,{\\mbox{$\\mu$m}} images \\citep{dale09},\n {\\it Herschel}\n\n\\citep{herschel,pacs} \n160\\,{\\mbox{$\\mu$m}} image \\citep{kingfish}, and\nNational Science Foundation's (NSF's) Karl G. Jansky Very Large Array (VLA) 3.1\\,GHz image \\citep{mulcahy17}.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nAccording to \\citet{kamphuis92}, M74 harbours an off-centre asymmetric {\\sc Hi} tail, located \non the south-western outskirts of the galaxy, outside the optical disc\n(Fig.~\\ref{fig:lowres}). The feature is detected over 20{\\arcmin} (55\\,kpc) and contains \n$\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=8.95$, or 7.5\\% of the total atomic gas of M74. {SN\\,2002ap} is located \nwhere this feature connects with the symmetric disc of M74.\nThe velocity pattern of the feature is irregular \n(Fig.~\\ref{fig:lowres}).\nThis gas does not follow the overall rotation of the gas disc, as evidenced by both negative and positive velocity residuals from the disc models at this location presented in Figs.~8 and 9 of \\citet{kamphuis92}. \n\nThe higher resolution THINGS data are not sensitive to such large scales, but allow detailed investigation of the local environments of SNe.\nAt this resolution, the position of {SN\\,2002ap} is not associated with any strong concentration of atomic or molecular gas (Fig.~\\ref{fig:image}).\n{SN\\,2002ap} is located $\\sim80\\arcsec$ (3.6\\,kpc) south-west of the main spiral arm running from the south to the west of the galaxy (marked as a curved region on Figs.~\\ref{fig:image} and \\ref{fig:image2}, traced clearly on all images up to the southernmost point, and by the {\\sc Hi} and 3.1\\,GHz images to the west) \nand $\\sim20\\arcsec$ (0.9\\,kpc) from \na bright {\\sc Hi} knot to the north. The main spiral arm in the south is also visible in the {\\sc Hi} velocity map in which \nthe integrated mean velocities show a larger deviation from the systemic velocity in the interarm regions.\n\nMoreover, {SN\\,2002ap} exploded away from regions of significant star formation activity and very little emission is present at its position at any wavelength (Fig.~\\ref{fig:image2}). There is, however, a faint star-forming region visible in the UV $\\sim4\\arcsec$ (180\\,pc) away from the SN position (this is not the object $10\\arcsec$ away mentioned by \\citealt{crowther13}).\n{SN\\,2002ap} is also outside the CO disc.\n\nThe other three type II SNe in M74 exploded along the most prominent spiral arm (running from the east to south of M74; curved regions on Figs.~\\ref{fig:image} and \\ref{fig:image2}), but are displaced from the arm towards the outside \nby $\\sim25\\arcsec$ ($\\sim1$\\,kpc). This is especially evident in the H$\\alpha$, $3.6\\,\\mbox{$\\mu$m},$ and CO images. SN\\,2013ej and 2019krl exploded in interarm regions with very little {\\sc Hi}, 8, 160\\,{\\mbox{$\\mu$m},} and 3.1\\,GHz emission.  All three type II SNe exploded in regions with undetectable CO emission.\n\nWe have investigated the large-scale environment of M74. Within 150\\,kpc (55{\\arcmin}) and $\\pm500\\,\\mbox{km\\,s$^{-1}$}$ from M74 (velocity of $627\\,\\mbox{km\\,s$^{-1}$}$), NED lists three galaxies:\nUGC\\,1171,\nUGC\\,1176,\nboth to the east,\nand the much fainter SDSS J013800.30+145858.1\nto the south. \nAll of them are detected in {\\sc Hi} by the Arecibo Legacy Fast ALFA Survey (ALFALFA; \\citealt{haynes18}). In Table~\\ref{tab:other} we list their properties.\n\nIn the ALFALFA catalogue within 200{\\arcsec} (540\\,kpc) of M74 there are in total 13 galaxies. All but one have atomic gas masses $7<\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})<9$. Only NGC\\,660, 426\\,kpc away, with $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=9.59$ has a mass comparable to that of M74.  \nThe positions of these galaxies are shown on Fig.~\\ref{fig:comp}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,clip]{companion.eps} \n\\end{center}\n\\caption{Large-scale environment around M74 within $\\pm500\\,\\mbox{km\\,s$^{-1}$}$. The width of the panel is 400{\\arcmin} ($\\sim1$\\,Mpc). Circles show the sizes of larger galaxies: 20{\\arcmin} for M74, 10{\\arcmin} for NGC\\,660, 5{\\arcmin} for UGC\\,1176, and 1.4{\\arcmin} for UGC\\,1171 (a small circle next to UGC\\,1176). Crosses show the positions of additional galaxies detected at the {\\sc Hi} line by ALFALFA \\citep{haynes18}. \n}\n\\label{fig:comp}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Properties of galaxies in the vicinity of M74 from the ALFALFA survey  \\citep{haynes18}.}\n    \\centering\n    \\begin{tabular}{lrccrccccc}\n    \\hline\\hline\n    Galaxy & ID & RA & Dec & \\multicolumn{2}{c}{Dist$_{\\rm M74}$} & $z$ & $V_{\\rm helio}$ & $f_{HI}$ & $\\log(M_{\\rm HI})$ \\\\ \n     & & (deg) & (deg) & ($'$) & (kpc) & & ($\\mbox{km\\,s$^{-1}$}$) & (Jy\\,$\\mbox{km\\,s$^{-1}$}$) & ($\\mbox{$M_\\odot$}$) \\\\\n     \\hline\n    M74 & 1149 & 24.18500 & 15.79222 & $\\cdots$ & $\\cdots$ & 0.002190 & 657 & $424.30\\pm0.18$ & $9.73\\pm0.1\\phantom{0}$ \\\\\n    UGC\\,1171 & 1171 & 24.93708 & 15.89611 & 44.6 & 120 & 0.002463 & 738 & $\\phantom{10}2.07\\pm0.05$ & $7.42\\pm0.09$ \\\\\n    UGC\\,1176 & 1176 &  25.03167 & 15.90167 &  50.6 & 137 & 0.002103 & 630 & $\\phantom{1}31.22\\pm0.06$ & $8.78\\pm0.1\\phantom{0}$ \\\\\n    SDSSJ013\n    & 112503 & 24.50583 & 14.99333 & 51.6 & 139 & 0.002478 & 743 & $\\phantom{10}0.56\\pm0.05$ & $7.14\\pm0.2\\phantom{0}$ \\\\\n    NGC\\,660 & 1201 & 25.76167 & 13.64000 &  157.9 & 426 & 0.002830 & 848 & $148.39\\pm0.14$ & $9.59\\pm0.34$ \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:other}\n    \\tablecomments{The columns show the galaxy name (that of SDSSJ013800.30+145858.1 has been abbreviated), ALFALFA ID, position, projected distance to M74, redshift, heliocentric velocity, {\\sc Hi} flux, and the atomic gas mass.}\n\\end{table*}\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe estimated mass of the progenitor of  {SN\\,2002ap} implies that it was formed 3.2--5.6\\,Myr before the explosion, whereas the progenitors of \nSN\\,2003gd and 2013ej\nwere formed 10--55\\,Myr before the explosions \n(assuming a main-sequence lifetime of $10^{10}\\,\\mbox{yr} \\times [M\/\\mbox{$M_\\odot$}\\mbox{$]$} ^{-2.5}$; \\citealt{kippenhahn90}).\nThis lifetime for {SN\\,2002ap} agrees with the single-progenitor estimate of \\citet[][5\\,Myr]{zapartas17}, but is lower than the binary-progenitor estimate (20\\,Myr). Similarly, \\citet{maund18} obtained an age of 15\\,Myr based on the analysis of stars in the vicinity of {SN\\,2002ap}.\n\n\\subsection{Type Ic-BL {SN\\,2002ap}}\n\n{The supernova SN\\,2002ap} is the fourth known explosion of a presumably massive star located close to a concentration of atomic gas. Similarly to {SN\\,2002ap},  GRB\\,060505 and SN\\,2009bb were both located close to {\\sc Hi} extensions, whereas GRB\\,980425 was located close to the most significant {\\sc Hi} concentration \\citep{michalowski15hi,michalowski18,arabsalmani15b,arabsalmani19}.\n\nWe can assess the statistical significance of the associations of the {\\sc Hi} off-centre concentrations with the GRB and SN positions by investigating the probability of four GRBs and SNe exploding by chance in the quadrants of their hosts at which these {\\sc Hi} concentrations are located. For a given explosion this probability is $0.25$, so for four out of four analysed cases this is $(0.25)^4\\sim0.004$. This corresponds to $\\sim3\\sigma$, so the associations are unlikely to be random, but a larger sample is needed to confirm this result.\n\nThe case of {SN\\,2002ap} adds to \nthe hypothesis put forward in \\citet{michalowski15hi} that the progenitors of these explosions are born when atomic gas is accreted from the intergalactic medium. Indeed, \\citet{kamphuis92} concluded that the south-western {\\sc Hi} extension in M74 is a result of an accretion event \nbecause it has not settled yet, and is inconsistent with the rotation of the gas disc. This extension may be the gas flowing in and feeding star formation directly, or distorting gas on the outskirts of the optical disc of the galaxy leading to star formation at the position of {SN\\,2002ap}.\n\nThe asymmetric tail in M74 has a mass of $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=8.95$. This is comparable to the mass of the most massive galaxy within 150\\,kpc  (UGC\\,1176) and only a factor of four less than the atomic gas mass of NGC\\,660.\nThe sum of atomic masses of all galaxies within 200{\\arcmin} (540\\,kpc) excluding NGC\\,660 is $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=9.36$.\nHence, the tail in M74 might have come from UGC\\,1176 if it was significantly distorted by the interaction and lost half of its gas, or from NGC\\,660.\nIt could also be a remnant of a galaxy similar to UGC\\,1176 that has been accreted entirely (as postulated by \\citealt{kamphuis92}), or a result of the accretion of intragroup medium.\n\nIn principle this tail could be a remnant of a  tidal feature created by interaction with these galaxies, but we found this interpretation unlikely. First, the tail does not resemble recent tidal features, whereas an older feature would  wind almost symmetrically around the galaxy. Second, simulations shows that tidal tails are created on both sides of interacting galaxies \\citep{hopkins06b,hayward12,hayward14,pettitt16,oh16}, whereas M74 does not have such feature on the other side.\nWe also note that the asymmetric nature of M74, with the southern arm being stronger than the northern one (Fig.~\\ref{fig:image}), could be a result of interaction with the UGC\\,1176\/1171 pair.\n\n\nThe only possible counter-example of a potential  explosion of a massive star without an associated gas concentration is the enigmatic transient AT\\,2018cow, whose host galaxy does not show such off-centre asymmetric {\\sc Hi} features  \\citep{michalowski19} and possibly only a gas ring \\citep{roychowdhury19}. \nSuch an {\\sc Hi} ring would also be apparent for M74 if it was further away so the sensitivity and  resolution were poorer, because the central part is devoid of atomic gas, likely due to conversion to the molecular phase (Fig.~\\ref{fig:image}).\nTo demonstrate this we smoothed the {\\sc Hi} VLA image with a Gaussian with a full width half maximum of 100{\\arcsec} (4.5\\,kpc;   Fig.~\\ref{fig:smooth} in the appendix). At this resolution the spiral structure of M74 resembles an irregular ring, similar to that detected for the AT\\,2018cow host (for which the resolution was around 2\\,kpc).\nHowever, the nature of AT\\,2018cow is not clear, so it may not be connected with the explosion of a massive star\n(\\citealt{prentice18,liu18,kuin19,perley19,soker19,lyutikov19,bietenholz20}, but see \\citealt{prentice18,riverasandoval18,margutti19,fox19,huang19}).\n\nIt is unlikely that the lack of molecular gas or star formation at the position of {SN\\,2002ap} is due to the progenitor being kicked out of a star-forming region. The velocities of runaway stars are up to 200\\,{\\mbox{km\\,s$^{-1}$}} \\citep{blaauw93,%\nhoogerwerf01,%\neldridge11%\n}, which corresponds to 1\\,kpc per 5\\,Myr. This is only $\\sim20\\arcsec$ at the distance of M74, so not sufficient to move the birth place of the {SN\\,2002ap} progenitor to any place of significant star formation or CO concentration.\nThis is true even if the lifetime of the progenitor is three to four times longer (15--20\\,My; \\citealt{zapartas17}, \\citealt{maund18}).\nCO deficiency at  GRB positions was also claimed by \\citet{hatsukade14}, \\citet{stanway15}, and \\citet{michalowski16}, but \\citet{perley17}, \\citet{michalowski18co}, and \\citet{arabsalmani18} suggest an alternative. If the lack of molecular gas is confirmed for a larger sample of type Ic-BL SNe, this would support the hypothesis of {\\sc Hi}-fuelled star formation giving rise to the birth of their progenitors \\citep{michalowski15hi}.\n\n\n\nIt is unlikely that the {SN\\,2002ap} progenitor moved to its explosion position due to a random kick.\nAssuming a lifetime of 5\\,Myr, the {SN\\,2002ap} progenitor could not be born in the main arm, as the required velocity is 700\\,{\\mbox{km\\,s$^{-1}$}} to cross 3.6\\,kpc. Even the closest bright {\\sc Hi} knot \nto the north\nis likely too far to be the birthplace, as this would require a velocity of 175\\,{\\mbox{km\\,s$^{-1}$}}. \nSuch  velocity kicks are at the high end for runaway stars \\citep{hoogerwerf01}. For the longer lifetime estimates of 15--20\\,Myr, the required velocities from the spiral arm would be 235--175\\,\\mbox{km\\,s$^{-1}$}, so still too high for the {SN\\,2002ap} progenitor to have been born there. However, in such a case it is feasible that it was born in the closest bright {\\sc Hi} knot to the north,\nas this would require velocities of 60--40\\,\\mbox{km\\,s$^{-1}$}.\n\n\\subsection{Type II SN\\,2003gd, 2013ej, and 2019krl}\n      \nAll type II SNe in M74 \nare not located close to the {\\sc Hi} extension, so are unlikely connected to gas accretion.\nThey are located at the outside edge of a spiral arm. This can be explained by either of two scenarios: \nby a gas density build-up and shock scenario at the edge of the arm, or by SN progenitors moving away from the arm during their lifetimes.\n\nThe first possibility is that the SN progenitors are born when gas is piling up and shocked at the edge of the arm when gas clouds are being swept up by the arm, as explained by the spiral density wave theory (\\citealt{shu16}). This is similar to the hypothesis presented in \\citet{michalowski14} that GRB progenitors are preferentially born in high-density gas.\nWe note that the amount of gas piling up at the edge of the arm giving rise to the birth of SN progenitors cannot be large, because the concentration is not detected with {\\sc Hi} or CO observations. \n\nThe SNe in M74 are on the outside of the spiral arm, so this scenario is only valid if they are outside the corotation radius (where the orbital velocity is equal to the spiral pattern speed), so the arm is catching up with gas that is moving slower \\citep{shu16}. \n\\citet{aramyan16} found that core-collapse SNe are indeed shifted towards the outside edge of spiral arms as long as they are  outside the corotation radius.\nUnfortunately the accuracy of the estimate of the corotation radius for M74 is not sufficient to test this. The corotation radius given by \\citet{scarano13}\\footnote{$4.6\\pm1.2$\\,kpc and their adopted scale of 1.95 kpc arcmin$^{-1}$.} is $(2.4\\pm0.6)\\arcmin$, corresponding to $(6.4\\pm1.7)\\,$kpc with our adopted distance, so it cannot be established whether type II SNe are inside or outside this radius. \n\\citet{karapetyan18} quoted a slightly lower (but consistent within errors) value of the corotation radius\\footnote{The ratio of the corotation and isophotal  ($R_{25}=5.52\\arcmin$) radii of  $0.34\\pm0.09$.} of $(1.9\\pm0.5)\\arcmin$, concluding that SNe\\,2003gd and 2013ej are  indeed outside the corotation radius, supporting the spiral density wave scenario for their formation.\nThis scenario cannot explain the birth of the {SN\\,2002ap} progenitor, because gas should not pile up so far away (3.6\\,kpc; Fig.~\\ref{fig:image}) from the spiral arm.\n\n\nHowever, the region of increased gas density and shocks, predicted by the spiral density wave theory, does manifest itself with increased star formation, and therefore more intense UV and H$\\alpha$ emission. These main sites of star formation are where SN progenitors should be born, not 1\\,kpc away. This scenario does not explain this discrepancy.\n\nThe second possibility is that SN progenitors are in fact born in the spiral arm, but, due to their orbital motions, move away before they explode. \nInside the corotation radius stars (and gas) are moving faster than the spiral pattern, so are constantly drifting towards the outside of the arm. \nThis means that SNe with progenitors with long enough lifetimes should be happening preferentially outside the arm. On the other hand, H$\\alpha$ emission is dominated by stars born very recently (i.e. after the SN progenitors), which have therefore not yet moved away from the arm.\n\nThe lifetimes of type II progenitors imply that they would need to have (reasonable) velocities of $18$--$100\\,\\mbox{km\\,s$^{-1}$}$ with respect to the arm to cross 1\\,kpc from the arm to their current position.\nWe note that this velocity is not due to any random-direction kick, but is the orbital velocity minus the spiral pattern speed.\n\n\n\n\n\nThe orbital migration scenario requires that type II SNe progenitors are inside the corotation radius, so their orbital motion is faster than the spiral pattern and so they can leave it on the outside edge \\citep[e.g.][]{aramyan16}.\nThis effect is also visible in M51 where younger stellar clusters have a distribution that has the strongest correlation with the distribution of star formation, and they are shifted towards the outside of the arm inside the corotation radius \\citep{scheepmaker09}.\n\nThis scenario cannot explain the position of {SN\\,2002ap}, because it is located securely outside the corotation radius, so cannot leave the spiral arm at the outside edge. Instead, as we discuss above, it may\nbe connected with the gas accretion visible in the {\\sc Hi} map of \\citet{kamphuis92}.\n\nThe position of SNe away from the main sites of star formation (spiral arms) is not unique to M74. We investigated galaxies with four or more core-collapse SNe and with well-separated spiral structures from the list of \\citet{thone09}. NGC\\,6946 and 4303 hosted nine and six type II SNe, respectively, and only one (1948B) and two (1999gn and 2006ov) exploded in spiral arms; the rest exploded in interarm regions or outside the detectable stellar disk (Figs.~9 and 11 of \\citealt{anderson13}). \nThis is consistent with the spiral density wave theory and the migration scenario described above.\nIndeed, larger samples of type II SNe show that they do not concentrate in the brightest regions of their hosts \\citep{fruchter06,anderson08,\nleloudas11}.\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nWe have used archival {\\sc Hi} and CO data for M74 (not previously investigated in the context of SN positions), together with H$\\alpha$ and continuum images.\n{SN\\,2002ap} is located  at  the end of  an  off-centre asymmetric 55\\,kpc-long {\\sc Hi} extension containing 7.5\\% of the total atomic gas of M74. \nIt is the fourth known explosion of a presumably massive star that is located close to the concentration of atomic gas (after GRB\\,980425, 060505, and SN\\,2009bb). \nIt is unlikely that all these associations are random (at a $3\\sigma$ significance), so\nthe case of {SN\\,2002ap} adds to\n the evidence that the birth of the progenitors of type Ic-BL SNe and GRBs is connected with the accretion of atomic gas from the intergalactic medium.\nThe {\\sc Hi} extension could come from tidally disrupted companions of M74, or be a remnant of a galaxy or a gas cloud in the intragroup medium accreted entirely. \n\nThe type II SNe in M74 do not seem to be related to gas accretion.\nThe fact that \nthey\nare located at the outside edge of a spiral arm suggests either that their progenitors are born when gas is piling up there, reaching high density, or that SN progenitors move away from the arm during their lifetimes, due to their orbital motions.\nThis is also similar for NGC\\,6946 and 4303, with eight out of nine and four out of six type II SNe, respectively, located in interarm regions or outside the detectable stellar discs, not in the spiral arms.\n\n\\begin{acknowledgements}\n\n\nWe wish to thank the referee for careful and important suggestions, Joanna Baradziej and Phillip Hopkins  for discussion and comments, and Frank Briggs for permission to use the figure from \\citet{kamphuis92}.\n\nM.J.M.~acknowledges the support of \nthe National Science Centre, Poland through the SONATA BIS grant 2018\/30\/E\/ST9\/00208, and of the Polish-U.S. Fulbright Commission.\nJ.H.~was supported by a VILLUM FONDEN Investigator grant (project number 16599).\nP.K.~is partially supported by the BMBF project 05A17PC2 for D-MeerKAT.\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThis work made use of HERACLES, `The HERA CO-Line Extragalactic Survey'.\nThis research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.\nThis work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.\n{\\it Herschel} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.\nThis research has made use of \nthe Open Supernova Catalog (\\url{https:\/\/sne.space});\nNASA\/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration;\nSAOImage DS9, developed by the Smithsonian Astrophysical Observatory \\citep{ds9};\nEdward Wright cosmology calculator \\citep{wrightcalc};\nthe WebPlotDigitizer of Ankit Rohatgi ({\\tt arohatgi.info\/WebPlotDigitizer});\nand NASA's Astrophysics Data System Bibliographic Services.\n\\end{acknowledgements}\n\n\n\\input{ms.bbl}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn \\emph{alternating sign matrix} (ASM) is a square matrix with entries in $\\{0,1,-1\\}$ such that in each row and each column the non-zero entries alternate and sum up to $1$. Robbins and Rumsey introduced alternating sign matrices in the 1980s \\cite{lambda}  when studying their \\emph{$\\lambda$-determinant} (a generalization of the classical determinant) and showing that the $\\lambda$-deter\\-mi\\-nant can be expressed as a sum over all alternating sign matrices of fixed size. The classical determinant is obtained from this by setting $\\lambda=-1$, in which case the sum reduces so that it extends only over all ASMs \\emph{without} $-1$'s, i.e., permutation matrices, and the well-known formula of Leibniz is recovered.\nNumerical experiments led Robbins and Rumsey to conjecture that the number of $n \\times n$ alternating sign matrices is given by the surprisingly simple product formula\n\\begin{equation}\n\\label{asm}\n\\prod_{i=0}^{n-1} \\frac{(3i+1)!}{(n+i)!}.\n\\end{equation}\n\n\\medskip\n\nBack then the surprise was even bigger when they learned from Stanley (see \\cite{BrePro99,Bre99}) that this product formula had recently also appeared in Andrews' paper \\cite{And79} on his proof of the weak Macdonald conjecture, which in turn provides a formula for the number of \\emph{cyclically symmetric plane partitions}. As a byproduct, Andrews had introduced \\emph{descending plane partitions}\u00a0and had proven that the number of descending plane partitions (DPPs) with parts at most $n$ is also equal to \\eqref{asm}. Since then the problem of finding an explicit bijection between alternating sign matrices and descending plane partitions has attracted considerable attention from combinatorialists and to many of them it is a miracle that such a bijection has not been found so far. All the more so because Mills, Robbins and Rumsey had also introduced several ``statistics'' on alternating sign matrices and on descending plane partitions for which they had strong numerical evidence that the joint distributions coincide as well, see \\cite{MilRobRum83}.\n\n\\medskip\n\nThere were a few further surprises yet to come. Robbins introduced a new operation on plane partitions, \\emph{complementation}, and had strong numerical evidence that totally symmetric self-complementary plane partitions (TSSCPPs) in a $2n \\times 2n \\times 2n$-box are also counted by \\eqref{asm}. Again this was further supported by statistics that have the same joint distribution as well as certain refinements, see \\cite{MilRobRum86,Kra96,krattsurvey,bianecheballah}. We still lack an explicit bijection between TSSCPPs and ASMs, as well as between TSSCPPs and DPPs.\n\n\\medskip\n\nIn his collection of bijective proof problems (which is available from his webpage) Stanley says the following about the problem of finding all these bijections: ``\\emph{This is one of the most intriguing open problems in the area of bijective proofs.}'' In Krattenthaler's survey on plane partitions \\cite{krattsurvey} he expresses his opinion by saying: ``\\emph{The greatest, still unsolved, mystery concerns the question of what plane partitions have to do with alternating sign matrices.}''\n\n\\medskip\n\nMany of the above mentioned conjectures have since been proved by non-bijective means: Zeilberger \\cite{Zei96a} was the first who proved that $n \\times n$ ASMs are counted by \\eqref{asm}. Kuperberg gave another shorter proof \\cite{Kup96}\u00a0based on the remarkable observation that the \\emph{six-vertex model} (which had been introduced by physicists several decades earlier) with domain wall boundary conditions is equivalent to ASMs, see \\cite{ElkKupLarPro92a,ElkKupLarPro92b}, and he used the techniques that had been developed by physicists to study this model. Andrews enumerated TSSCPPs in \\cite{And94}. The equivalence of certain statistics for ASMs and of certain statistics for DPPs has been proved in \\cite{BehDifZin12,BehDifZin13}, while for ASMs and TSSCPPs see \\cite{Zei96b,FonZin08}, and note in particular that already in Zeilberger's first ASM paper \\cite{Zei96a} he could deal with an important refinement.\nFurther work including the study of \\emph{symmetry classes} has been accomplished; for a more detailed description of this we defer to \\cite{BehFisKon17}. Then, in very recent work, alternating sign triangles (ASTs) were introduced in \\cite{AyyBehFis16}, which establishes a fourth class of objects that are equinumerous with ASMs, and also in this case nobody has so far been able to construct a bijection.\n\n\\medskip\n\nAnother aspect that should be mentioned here is Okada's work \\cite{Oka06}\u00a0(see also \\cite{stro}), which hints at a connection between ASMs and representation theory that has not yet been well understood. He observed that a certain multivariate generating function (a specialization at a root of unity of the partition function that had been introduced by physicists in their study of the six-vertex model) can be expressed---up to a power of $3$---by a single \\emph{Schur polynomial}. Since Schur polynomials are generating functions of semistandard tableaux, this establishes yet another challenging open problem for combinatorialists inclined to find bijections.\n\n\\medskip\n\nThe proofs of the results briefly reviewed above contain rather long and complicated computations, and include hardly any arguments of a combinatorial flavor. In fact it seems that all ASM-related identities for which there exists a bijective proofs are trivial, with the exception of the rotational invariance of fully packed loop configurations. This was proved by Wieland  \\cite{wieland} bijectively and is also used in the celebrated proof of the Razumov-Stroganov (ex-)conjecture \\cite{razstrog}.\n\n\n\\medskip\n\nWe come now to the purpose of the current paper. This is the first paper in a planned series that seeks to give the first bijective proofs of several results described so far. The seed of the idea to do so came from a brief discussion of the first author with Zeilberger on the problem of finding such bijections at the AMS-MAA Joint Mathematics Meetings 2019. Zeilberger mentioned that such bijections can be constructed from existing ``computational'' proofs, however, most likely these bijections are complicated. The authors of the current paper agree, in fact the first author gave her ``own'' proof of the ASM theorem in \\cite{Fis06,Fis07,Fis16} and expressed some speculations in this direction in the final section of the last paper. It is also not implausible that a simple satisfactory bijective proof of the ASM theorem does not exist at all. Combinatorialists have failed to find such bijections for decades now, and we may start to ask ourselves why we are not rewarded for these efforts.\n\n\\medskip\n\nThis is how the authors of the current paper decided to work on converting the proof in \\cite{Fis16} into a bijective proof. After having figured out how to actually convert computations and also having shaped certain useful fundamental concepts related to  \\emph{signed sets} (see Section~\\ref{sec:ss}), the translation of several steps became quite straightforward; some steps were quite challenging. Then a certain type of (exciting) dynamics evolved, where the combina\\-to\\-rial point of view led to simplifications and other modifications, and after this process the original ``computational'' proof is in fact rather difficult to recognize. For several obvious reasons, we find it essential to check all our constructions with computer code; to name one it can possibly be used to identify new equivalent statistics.\n\n\\medskip\n\nAfter the above mentioned simplifications, it seems that \\emph{signs} seem to be unavoidable. After all, if there would be a simple bijective proof that avoided signs, would it not also be plausible that such a proof could be converted into a simple ``computational'' proof that avoids signs? Such a proof has also not been found so far.\n\n\\medskip\n\nIn the remainder of the introduction we discuss the result that is proved bijectively in this paper, in particular we discuss why signed enumerations seem to be unavoidable from this point of view. We also sketch a few ideas informally before giving rigorous definitions and proofs later on.\n\n\\subsection*{The operator formula}\u00a0\nWe use the well-known correspondence between order $n \\times n$ ASMs and \\emph{monotone triangles} with bottom row $1,2,\\ldots,n$. A \\emph{monotone triangle} is a triangular array $(a_{i,j})_{1 \\le j \\le i \\le n}$ of integers, where the elements are usually arranged as follows\n\\begin{equation}\n\\label{triangle}\n\\begin{array}{ccccccccccccccccc}\n  &   &   &   &   &   &   &   & a_{1,1} &   &   &   &   &   &   &   & \\\\\n  &   &   &   &   &   &   & a_{2,1} &   & a_{2,2} &   &   &   &   &   &   & \\\\\n  &   &   &   &   &   & \\dots &   & \\dots &   & \\dots &   &   &   &   &   & \\\\\n  &   &   &   &   & a_{n-2,1} &   & \\dots &   & \\dots &   & a_{n-2,n-2} &   &   &   &   & \\\\\n  &   &   &   & a_{n-1,1} &   & a_{n-1,2} &  &   \\dots &   & \\dots   &  & a_{n-1,n-1}  &   &   &   & \\\\\n  &   &   & a_{n,1} &   & a_{n,2} &   & a_{n,3} &   & \\dots &   & \\dots &   & a_{n,n} &   &   &\n\\end{array},\n\\end{equation}\nsuch that the integers increase weakly along $\\nearrow$-diagonals and $\\searrow$-diagonals, and increase strictly along rows, i.e.,\n$a_{i,j} \\le a_{i-1,j} \\le a_{i,j+1}$ and $a_{i,j} < a_{i,j+1}$  for all $i,j$ with $1 \\le j < i \\le n$. In order to convert an ASM into the corresponding monotone triangle, add to each entry all the entries that are in the same column above it, and record then row by row the positions of the $1$'s, see Figure~\\ref{ASM-MT} for an example.\n\n\n\\begin{figure}\n$$\n\\left(\n\\begin{matrix}\n0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 1 & 0 & -1 & 1 & 0 \\\\\n1 & -1 & 0 & 1 & -1 & 1 \\\\\n0 & 1  & 0 & -1 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0\n\\end{matrix} \\right) \\rightarrow\n\\left(\n\\begin{matrix}\n0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 1 & 0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 1 & 0 & 1 \\\\\n1 & 1  & 0 &0 & 1 & 1 \\\\\n1 & 1 & 0 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1\n\\end{matrix} \\right) \\rightarrow\n\\begin{array}{ccccccccccc}\n  & & & & & 4  & & & & &  \\\\\n  & & & & 2 &  & 5  & & & &  \\\\\n  & & & 1 &  & 4 &   & 6 & & &  \\\\\n & & 1 &  & 2  &  & 5  &  & 6 & &  \\\\\n &1  &  & 2 &   & 4  &   & 5  &  & 6 &  \\\\\n1  &  & 2 &  & 3  &   & 4   &   & 5  &  & 6\n\\end{array}\n$$\n\\caption{\\label{ASM-MT} ASM $\\rightarrow$ partial columnsums $\\rightarrow$ monotone triangle}\n\\end{figure}\n\n\\medskip\n\nThe following \\emph{operator formula}  for the number of monotone triangles with prescribed bottom row was first proved in \\cite{Fis06} (see \\cite{Fis10,Fis16} for simplifications and generali\\-za\\-tions). Note that we allow arbitrary strictly increasing bottom rows.\n\n\\begin{theorem}\n\\label{operator}\nLet $k_1 < k_2 < \\ldots < k_n$ be a sequence of strictly increasing integers. The number of monotone triangles with bottom row $k_1,\\ldots,k_n$ is\n\\begin{equation}\n\\label{operatorexpr}\n\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right) \\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i},\n\\end{equation}\nwhere ${\\operatorname{E}}_x$ denotes the shift operator, i.e., ${\\operatorname{E}}_x p(x) = p(x+1)$.\\footnote{The formula has to be understood as follows: Take $\\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i}$ and treat the $k_i$'s as indeterminates. Apply $\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right)$ to this polynomial to obtain another polynomial. Only then the $k_i$'s can be specialized to the actual values.}\n\\end{theorem}\n\n\n\nThe purpose of this paper is to provide a bijective proof of Theorem~\\ref{operator}. While the operator formula is an interesting result in its own right, it has also been the main tool for proofs of several results mentioned above. This will be reviewed in the final section of this paper along with indications for future projects on converting also these proofs into bijective proofs.\n\n\\medskip\n\nIn order to be able to construct a bijective proof of Theorem~\\ref{operator}, we need to interpret \\eqref{operatorexpr} combinatorially. Recall that \\emph{Gelfand-Tsetlin patterns} are defined as monotone triangles with the condition on the strict increase along rows being dropped, see  \\cite[p.\\ 313]{Sta99} or \\cite[(3)]{gelfand} for the original reference\\footnote{Gelfand-Tsetlin patterns with bottom row $0 \\le k_1 \\le k_2 \\le \\ldots \\le k_n$ are in an easy bijective correspondence with seminstandard tableaux of shape $(k_n,k_{n-1},\\ldots,k_1)$ and entries in $\\{1,2,\\ldots,n\\}$.}. It is well known that the number of Gelfand-Tsetlin patterns with bottom row $k_1 \\le k_2 \\le \\ldots \\le k_n$ is\n\\begin{equation}\n\\label{gelfandtsetlin}\n\\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i},\n\\end{equation}\nwhich is the operand in the operator formula \\eqref{operatorexpr}. Expanding $\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right)$ into $3^{\\binom{n}{2}}$ monomials in ${\\operatorname{E}}^{\\pm 1}_{k_1}, {\\operatorname{E}}^{\\pm 1}_{k_2}, \\ldots, {\\operatorname{E}}^{\\pm 1}_{k_n}$ (keeping a copy for each multiplicity), \\eqref{operatorexpr}\u00a0is a signed enumeration of certain Gelfand-Tsetlin patterns, where each monomial causes a deformation of the bottom row $k_1,\\ldots,k_n$. It is useful to encode these deformations by \\emph{arrow patterns} as defined in Section~\\ref{sec:recursion}, where we choose $\\swarrow$ if we pick ${\\operatorname{E}}_{k_p}$ from ${\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$, we choose $\\searrow$ if we pick ${\\operatorname{E}}_{k_q}^{-1}$, while we choose $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if we pick $-{\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$. Arranging the $\\binom{n}{2}$ arrows in a triangular manner so that the arrows coming from ${\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$ are situated in the $p$-th $\\swarrow$-diagonal and the $q$-th $\\searrow$-diagonal, and placing $k_1,\\ldots,k_n$ in the bottom row will allow us to describe the deformation coming from a particular monomial in a convenient way. The combinatorial objects associated with \\eqref{operatorexpr} then consist of a pair of such an arrow pattern and a Gelfand-Tsetlin pattern where the bottom row is a deformation of $k_1,\\ldots,k_n$ as prescribed by the arrow pattern. This will lead directly to the definition of \\emph{shifted Gelfand-Tsetlin patterns}.\n\n\\medskip\n\nA sign comes from picking $- {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$, but there is also a more subtle appearance. The deformation induced by the arrow pattern can cause a deformation of the increasing bottom row $k_1,k_2,\\ldots,k_n$  into a sequence that is not increasing. Therefore we are in need of an extension of the combinatorial interpretation of \\eqref{gelfandtsetlin} to any sequence $k_1,\\ldots,k_n$ of integers. Such an interpretation was given in \\cite{Fis05} and is repeated below in Section~\\ref{sec:gt}.\n\n\\subsection*{Outline of the bijective proof} Given a sequence $k_1 < \\ldots < k_n$, it suffices to find an injective map from the set of monotone triangles with bottom row $k_1,\\ldots,k_n$ to our shifted Gelfand-Tsetlin patterns associated with $k_1,\\ldots,k_n$ so that the images under this map have positive signs, along with a sign-reversing involution on the set of shifted Gelfand-Tsetlin patterns that are not the image of a monotone triangle.\n\n\\medskip\n\nWe will accomplish something more general, as we will also consider an extension of monotone triangles to all integer sequences $k_1,\\ldots,k_n$, see Section~\\ref{sec:recursion}, along with a sign function on these objects, and prove that the operator formula also holds in this instance. To do that, we will construct a sign-reversing involution on a subset of monotone triangles, another sign-reversing involution on a subset of shifted Gelfand-Tsetlin patterns, and a sign-\\emph{preser\\-ving} bijection between the remaining monotone triangles and the remaining shifted Gelfand-Tsetlin patterns. Note that this is actually equivalent to the construction of a bijection between the (disjoint) union of the ``positive'' monotone triangles and the ``negative'' shifted Gelfand-Tsetlin patterns, and the (disjoint) union of the ``negative'' monotone triangles and the ``positive'' shifted Gelfand-Tsetlin patterns. We call such maps \\emph{sijections} for general signed sets.\n\n\\medskip\n\nThe actual construction here will make use of the recursion underlying monotone triangles. For a monotone triangle with bottom row $k_1,\\ldots,k_n$, the eligible penultimate rows $l_1,\\ldots,l_{n-1}$ are those with\n$$\nk_1 \\le l_1 \\le k_2 \\le l_2 \\le \\ldots \\le l_{n-1} \\le k_n,\n$$\nand $l_1 < l_2 < \\ldots < l_{n-1}$. This establishes a recursion that can be used to construct all monotone triangles. Phrased differently, ``at'' each $k_i$ we need to sum over all $l_{i-1},l_i$ such that $l_{i-1}\u00a0\\le k_i \\le l_i$ and $l_{i-1} < l_{i}$.\\footnote{The degenerate cases $k_1$ and $k_n$ are slightly different.} However, we can split this into the following three cases:\n\\begin{enumerate}\n\\item Consider all $l_{i-1},l_i$ with $l_{i-1} < k_i \\le  l_i$.\n\\item Consider all $l_{i-1},l_i$ with $l_{i-1} \\le  k_i < l_i$.\n\\item Combining (1) and (2), we have done some double counting, thus we need to subtract the intersection, i.e., all $l_{i-1},l_i$ with $l_{i-1} < k_i < l_i$.\n\\end{enumerate}\nThis can be written as a recursion. The \\emph{arrow rows} in Section~\\ref{sec:recursion} are used to describe this recursion: we choose $\\nwarrow$ ``at'' $k_i$ if we are in Case (1), $\\nearrow$ in Case (2), and $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in Case (3). Our main effort will be to show ``sijectively'' that shifted Gelfand-Tsetlin patterns also fulfill this recursion.\n\n\\subsection*{Outline of the paper} The remainder of this paper is devoted to the bijective proof of Theorem~\\ref{operator} (or rather, the more general version with the increasing condition on $k_1,\\ldots,k_n$ dropped). In Section~\\ref{sec:ss} we lay the groundwork by defining concepts like signed sets and sijections, and we extend known concepts such as disjoint union, Cartesian product and composition for ordinary sets and bijections to signed sets and sijections. The composition of sijections will use a variation of the well-known Garsia-Milne involution principle \\cite{GarsiaMilne2,GarsiaMilne1}. Many of the signed sets we will be considering are signed boxes (Cartesian products of signed intervals) or at least involve them, and we define some sijections on them in Section~\\ref{sec:sb}. These sijections will be the building blocks of our bijective proof later on. In\nSection~\\ref{sec:gt} we introduce the extended Gelfand-Tsetlin patterns and construct some related sijections. In Section~\\ref{sec:recursion}, we finally define the extended monotone triangles as well as the shifted Gelfand-Tsetlin patterns (i.e., the combinatorial interpretation of \\eqref{operatorexpr}), and use all the preparation to construct the sijection between monotone triangles and shifted Gelfand-Tsetlin patterns. In the final section, we discuss further projects.\n\n\\medskip\n\nTo emphasize that we are not merely interested in the fact that two signed sets have the same size, but want to use the constructed signed bijection later on, we will be using a convention that is slightly unorthodox in our field. Instead of listing out results as lemmas and theorems with their corresponding proofs, we will be using the Problem--Construction terminology. See for instance \\cite{Voevodsky} and \\cite{Bauer}.\n\n\\section{Signed sets and sijections} \\label{sec:ss}\n\n\\subsection*{Signed sets}\n\nA \\emph{signed set} is a pair of disjoint finite sets: $\\u S = (S^+,S^-)$ with $S^+ \\cap S^- = \\emptyset$. Equivalently, a signed set is a finite set $S$ together with a sign function  $\\sign \\colon S \\to \\{1,-1\\}$. While we will mostly avoid the use of the sign function altogether (with the exception of monotone triangles defined in Section \\ref{sec:recursion}), it is useful to keep this description at the back of one's mind. Note that throughout the paper, signed sets are underlined. We will write $i \\in \\u S$ to mean $i \\in S^+ \\cup S^-$.\n\n\\medskip\n\nThe \\emph{size} of a signed set $\\u S$ is $|\\u S| = |S^+| - |S^-|$. The \\emph{opposite} signed set of $\\u S$ is $- \\u S = (S^-,S^+)$. We have $|{- \\u S}  | = -|\\u S|$. The \\emph{Cartesian product} of signed sets $\\u S$ and $\\u T$ is\n$$\\u S \\times \\u T = (S^+ \\times T^+ \\cup S^- \\times T^-,S^+ \\times T^- \\cup S^- \\times T^+),$$\nand we can similarly (or recursively) define the Cartesian product of a finite number of signed sets. We have\n$$|\\u S \\times \\u T| = |S^+| \\cdot |T^+| + |S^-| \\cdot |T^-| - |S^+| \\cdot |T^-| - |S^-| \\cdot |T^+| = |\\u S| \\cdot |\\u T|.$$\n\nThe intersection of signed sets $\\u S$ and $\\u T$ is defined as $\\u S \\cap \\u T = (S^+ \\cap T^+, S^- \\cap T^-)$, while the union $\\u S \\cup \\u T = (S^+ \\cup T^+, S^- \\cup T^-)$ is only defined when $S^+ \\cap T^- = S^- \\cap T^+ = \\emptyset$. Again, we can extend these definitions to a finite family of signed sets.\n\n\\begin{example}\n One of the crucial signed sets is the \\emph{signed interval}\n $$\\u{[a,b]} = \\begin{cases} ([a,b],\\emptyset) & \\mbox{if } a \\leq b \\\\ (\\emptyset,[b+1,a-1]) & \\mbox{if } a > b \\end{cases}$$\n for $a,b \\in {\\mathbb Z}$, {where $[a,b]$ stands for an interval in $\\mathbb{Z}$ in the usual sense}. We have $\\si{b+1}{a-1} = -\\si a b$ and $|\\u{[a,b]}| = b - a + 1$.\\\\\n We will also see many \\emph{signed boxes}, Cartesian products of signed intervals. Note that $S^+ = \\emptyset$ or $S^- = \\emptyset$ for every signed box $\\u S$.\n\\end{example}\n\n{Signed subsets $\\u T \\subseteq \\u S$ are defined in an obvious manner, in particular, for $s \\in \\u S$, we have}\n$$\\u{\\{s\\}} = \\begin{cases} (\\{s\\},\\emptyset) & \\mbox{if } s \\in S^+ \\\\ (\\emptyset,\\{s\\}) & \\mbox{if } s \\in S^- \\end{cases}.$$\n\nThe \\emph{disjoint union} of signed sets $\\u S$ and $\\u T$ is the signed set\n$$\\u S \\sqcup \\u T = (\\u S \\times (\\{0\\},\\emptyset)) \\cup (\\u T \\times (\\{1\\},\\emptyset))$$\nwith elements $(s,0)$ for $s \\in \\u S$ and $(t,1)$ for $t \\in \\u T$. If $\\u S$ and $\\u T$ are signed sets with $(S^+ \\cup S^-) \\cap (T^+ \\cup T^-) = \\emptyset$, we can identify $\\u S \\cup \\u T$ and $\\u S \\sqcup \\u T$.\n\n\\medskip\n\nMore generally, we can define the disjoint union of a family of signed sets $\\u S_t$, where the family is indexed with a signed set $\\u T$:\n$$\\bigsqcup_{t \\in \\u T} \\u S_t = \\bigcup_{t \\in \\u T} (\\u S_t \\times \\u{\\{t\\}}).$$\n{We get $\\bigsqcup_{t \\in \\u{[0,1]}} \\u S_t = S_0 \\sqcup S_1$. For $a,b \\in \\mathbb{Z}$, we may also write $\\bigsqcup_{i=a}^b \\u S_i$ instead of $\\bigsqcup_{i \\in \\si a b} \\u S_i$.}\n{As for the size, we have $$\\left| \\bigsqcup_{t \\in \\u T} \\u S_t  \\right|= \\sum_{t \\in T} | \\u{S_t} | \\cdot\n| \\u { \\{ t \\} } |.$$}\n\n\\medskip\n\nThe usual properties such as {associativity $(\\u S \\sqcup \\u T) \\sqcup \\u U = \\u S \\sqcup (\\u T \\sqcup \\u U)$ and} {distributivity}\n$(\\u S \\sqcup \\u T) \\times \\u U = \\u S \\times \\u U \\sqcup \\u T \\times \\u U$ also hold. {Strictly speaking, the $=$ sign here and sometimes later on indicates that there is an obvious and natural sign-preserving bijection between the two signed sets.} We summarize a few more basic properties that will be needed in the following and that are easy to prove.\n\\begin{enumerate}\n\\item $$\\bigsqcup\\limits_{\\q l \\in \\si{a_1}{b_1} \\times \\ldots  \\times \\si{a_{n}}{b_{n}}} \\u S_{l_1+c_1,\\ldots,l_n+c_n}  =\n\\bigsqcup\\limits_{\\q l \\in \\si{a_1+c_1}{b_1+c_1} \\times \\ldots  \\times \\si{a_{n}+c_{n}}{b_{n}+c_{n}}} \\u S_{l_1,\\ldots,l_n}$$\n\\item $$\n\\bigsqcup_{t \\in \\u T} \\bigsqcup_{u \\in \\u U} \\u S_{t,u} = \\bigsqcup_{(u,t) \\in \\u U \\times \\u T}\u00a0\n\\u S_{t,u} =    \\bigsqcup_{(t,u) \\in \\u T \\times \\u U}\u00a0\\u S_{t,u} =\n\\bigsqcup_{u \\in \\u U} \\bigsqcup_{t \\in \\u T} \\u S_{t,u}\n$$\n\\item\n$$\n\\bigsqcup_{t \\in \\bigsqcup_{u \\in \\u U} \\u T_u} \\u S_t = \\bigsqcup_{u \\in \\u U} \\bigsqcup_{t \\in \\u T_u} \\u S_t\n$$\n\\item $$\n- \\bigsqcup_{t \\in \\u T} \\u S_t  = \\bigsqcup_{t \\in \\u T} - \\u S_t= \\bigsqcup_{t \\in -\\u T} \\u S_t.\n$$\n\\end{enumerate}\n\n\\medskip\n\n\\subsection*{Sijections}\n\nThe role of bijections for signed sets is played by ``signed bijections'', which we call \\emph{sijections}. A sijection $\\varphi$ from $\\u S$ to $\\u T$,\n$$\\varphi \\colon \\u S \\Rightarrow \\u T,$$\nis an involution on the set $(S^+ \\cup S^-) \\sqcup (T^+ \\cup T^-)$ with the property $\\varphi(S^+ \\sqcup T^-) = S^- \\sqcup T^+$, {where $\\sqcup$ refers to the disjoint union for ordinary (``unsigned'') sets.}\n It follows that also $\\varphi(S^- \\sqcup T^+) = S^+ \\sqcup T^-$. {There is an obvious sijection $\\id_{\\u S} \\colon \\u S \\Rightarrow \\u S$.}\n\\medskip\n\nWe can think of a sijection as a collection of a sign-reversing involution on a subset of $\\u S$, a sign-reversing involution on a subset of $\\u T$, and a {sign-preserving} matching between the remaining elements of $\\u S$ with the remaining elements of $\\u T$. When $S^- = T^- = \\emptyset$, the signed sets can be identified with ordinary sets, and a sijection in this case is simply a bijection.\n\n\\medskip\n\n{A sijection is a  manifestation of the fact that two signed sets have the same size. Indeed,}\nif there exists a sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$, we have $|S^+| + |T^-| = |S^+ \\sqcup T^-| = |S^- \\sqcup T^+| = |S^-| + |T^+|$ and therefore $|\\u S| = |S^+| - |S^-| = |T^+| - |T^-| = |\\u T|$. A sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$ has an inverse $\\varphi^{-1} \\colon \\u T \\Rightarrow \\u S$ that we obtain by identifying $(T^+ \\cup T^-) \\sqcup (S^+ \\cup S^-)$ with $(S^+ \\cup S^-) \\sqcup (T^+ \\cup T^-)$.\n\n\\medskip\n\n\\comment{A sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$ is \\emph{simple} if  $\\varphi(S^+) = T^+$ and  $\\varphi(S^-) = T^-$.}\n\n\\medskip\n\nFor a signed set $\\u S$, there is a natural sijection $\\varphi$ from $\\u S \\sqcup (- \\u S)\n$\nto the empty signed set $\\u \\emptyset = (\\emptyset,\\emptyset)$. Indeed, the {involution} should be defined on {$(S^+ \\times \\{0\\} \\cup S^- \\times\\{1\\}) \\cup (S^- \\times \\{0\\} \\cup S^+ \\times \\{1\\})$ and map $S^+ \\times \\{0\\} \\cup S^- \\times\\{1\\}$ to $S^+ \\times \\{1\\} \\cup S^- \\times \\{0\\}$}, and {so} we can take $\\varphi((s,0),0) = ((s,1),0)$, $\\varphi((s,1),0) = ((s,0),0)$. {Note that in general, a sijection from a signed set $\\u S$ to $\\u \\emptyset$ is simply a sign-reversing involution on $\\u S$, in other words, a bijection between $S^+$ and $S^-$.} \\comment{In other words, a sijection $\\varphi \\colon \\u S \\Rightarrow \\u \\emptyset$ is equivalent to a simple sijection $\\varphi \\colon \\u S \\Rightarrow - \\u S$.}\n\n\\medskip\n\n{If we have a sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$, there is a natural sijection $-\\varphi \\colon -\\u S \\Rightarrow -\\u T$ (as a map, it is actually precisely the same).}\n\n\\medskip\n\n{If we have sijections $\\varphi_i \\colon \\u S_i \\Rightarrow \\u T_i$ for $i=0,1$, then there is a natural sijection $\\varphi \\colon \\u S_0 \\sqcup \\u S_1 \\Rightarrow \\u T_0 \\sqcup \\u T_1$.}\nMore interesting ways to create new sijections are described below in Proposition \\ref{prop:sijections}, but we will need this in our first construction {for the special case $\\u S_0 = \\u T_0$ and $\\varphi_0 = \\id_{\\u S_0}$.}\n\n\\medskip\n\nTo motivate our first result, note that if $a \\leq b \\leq c$ or $c < b < a$, then $\\si a c = \\si a b \\cup \\si{b+1}c= \\si a b \\sqcup \\si{b+1}c$. Of course, this does not hold in general; for $a = 1$, $b = 8$, $c = 5$, we have $\\si 1 5 = (\\{1,2,3,4,5\\},\\emptyset)$, $(\\si 1 8 \\sqcup \\si 9 5)^+ = (\\{(1,0),(2,0),(3,0),(4,0),(5,0),(6,0),(7,0),(8,0)\\}$ and $(\\si 1 8 \\sqcup \\si 9 5)^- = \\{(6,1),(7,1),(8,1)\\})$. The following, however, tells us that there is in general a sijection between {$\\si a c$ and\n$\\si a b \\sqcup \\si{b+1}c$.}\nThis map will be the crucial building block for more complicated sijections.\n\n\\begin{problem} \\label{prob:alpha}\n Given $a,b,c \\in {\\mathbb Z}$, construct a sijection\n $$\\alpha = \\alpha_{a,b,c} \\colon \\si a c \\Rightarrow \\si a b \\sqcup \\si{b+1}c.$$\n\\end{problem}\n\\begin{proof}[Construction]\n For $a \\leq b \\leq c$ and $c < b < a$, there is nothing to prove. For, say, $a \\leq c < b$, we have\n $$\\si a b \\sqcup \\si{b+1}c = (\\si a c \\sqcup \\si{c+1} b) \\sqcup \\si{b+1}c = \\si a c \\sqcup (\\si{c+1} b \\sqcup (-\\si {c+1}{b}))$$\n and since there is a sijection $\\si{c+1} b \\sqcup (-\\si {c+1}{b}) \\Rightarrow \\u \\emptyset$, we get a sijection $\\si a b \\sqcup \\si{b+1}c \\Rightarrow \\si a c$. The cases $b < a \\leq c$, $b \\leq c < a$, and $c < a \\leq b$ are analogous. \\comment{Note that in all cases, $\\si a b$ and  $\\si{b+1}c$ are either disjoint or one set is contained in the other.}\n\\end{proof}\n\n\\medskip\n\nThe following proposition describes composition, Cartesian product, and disjoint union of sijections. The composition is a variant of the well-known Garsia-Milne involution principle. All the statements are easy to prove{, and the proofs are left to the reader.}\n\n\\begin{prop} \\leavevmode \\label{prop:sijections}\n \\begin{enumerate}\n  \\item (Composition) Suppose that we have sijections $\\varphi \\colon \\u S \\Rightarrow \\u T$ and $\\psi \\colon \\u T \\Rightarrow \\u U$. For $s \\in \\u S$ (resp.\\ $u \\in \\u U$), define $\\psi \\circ \\varphi(s)$ {(resp.\\ $\\psi \\circ \\varphi(u)$)} as the last well-defined element in the sequence $s, \\varphi(s), \\psi(\\varphi(s)), \\varphi(\\psi(\\varphi(s))),\\ldots$ (resp.\\ $u, \\psi(u), \\varphi(\\psi(u)), \\psi(\\varphi(\\psi(u))),\\ldots$). Then $\\psi \\circ \\varphi$ is a well-defined sijection from $\\u S$ to $\\u U$. \\comment{If $\\varphi$ and $\\psi$ are simple, so is $\\psi \\circ \\varphi$.}\n  \\item (Cartesian product) Suppose we have sijections $\\varphi_i \\colon \\u S_i \\Rightarrow \\u T_i$, $i = 1,\\ldots,k$. Then $\\varphi = \\varphi_1 \\times \\cdots \\times \\varphi_k$, defined by\n$$\\varphi(s_1,\\ldots,s_k) = \\begin{cases} (\\varphi_1(s_1),\\ldots,\\varphi_k(s_k)) & \\mbox{if } \\varphi_i(s_i) \\in \\u T_i \\mbox{ for } i=1,\\ldots,k \\\\ (s_1,\\ldots,s_{j-1},\\varphi_j(s_j),s_{j+1},\\ldots,s_k) & \\mbox{if } \\varphi_j(s_j) \\in \\u S_j, \\varphi_i(s_i) \\in \\u T_i \\mbox{ for } i < j \\end{cases}$$\n{if $(s_1,\\ldots,s_k) \\in \\u S_1 \\times  \\dots \\times \\u S_k$ and\n$$\\varphi(t_1,\\ldots,t_k) = \\begin{cases} (\\varphi_1(t_1),\\ldots,\\varphi_k(t_k)) & \\mbox{if } \\varphi_i(t_i) \\in \\u S_i \\mbox{ for } i=1,\\ldots,k \\\\ (t_1,\\ldots,t_{j-1},\\varphi_j(t_j),t_{j+1},\\ldots,t_k) & \\mbox{if } \\varphi_j(t_j) \\in \\u T_j, \\varphi_i(t_i) \\in \\u S_i \\mbox{ for } i < j \\end{cases}$$\nif $(t_1,\\ldots,t_k) \\in \\u T_1  \\times \\dots \\times \\u T_k$,}\n  is a well-defined sijection from $\\u S_1 \\times \\cdots \\times \\u S_k$ to $\\u T_1 \\times \\cdots \\times \\u T_k$. \\comment{If $\\varphi_i$ are all simple sijections, so is $\\varphi$.}\n  \\item (Disjoint union) {Suppose we have signed sets $\\u T, \\u{\\widetilde T}$ and a sijection $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$. Further\\-more, suppose that for every $t \\in \\u T \\sqcup \\u{\\widetilde T}$, we have a signed set $\\u S_t$ and a sijection $\\varphi_t \\colon \\u S_t \\Rightarrow \\u S_{\\psi(t)}$ satisfying $\\varphi_{\\psi(t)} = \\varphi_t^{-1}$.  Then $\\varphi = \\bigsqcup_{t \\in \\u T \\sqcup \\u{\\widetilde T}} \\varphi_t$, defined by\n$$\\varphi(s_t,t) = \\begin{cases}\n(\\varphi_t(s_t),t) & \\mbox{if } t \\in \\u T \\sqcup \\u {\\widetilde T}, s_t \\in \\u S_t, \\varphi_t(s_t) \\in \\u S_t \\\\\n(\\varphi_t(s_t),\\psi(t)) & \\mbox{if } t \\in \\u T \\sqcup \\u {\\widetilde T}, s_t \\in \\u S_t, \\varphi_t(s_t) \\in \\u {S}_{\\psi(t)} \\\\\n\\end{cases}\n$$\nis a sijection $\\bigsqcup_{t \\in \\u T} \\u S_t \\Rightarrow \\bigsqcup_{t \\in \\u {\\widetilde T}} \\u {\\widetilde S}_t$.} \\comment{If $\\psi$ and $\\varphi_t$ are all simple sijections, so is $\\varphi$.}\n\\end{enumerate}\n\\end{prop}\n\n\n{One} important special case of Proposition \\ref{prop:sijections} (3) is $\\u T =  \\u {\\widetilde T}$ and $\\psi = \\id$. We have two sets of signed sets indexed by $\\u T$, $\\u S_{(t,0)} =: \\u{S}^{{0}}_t$ and $\\u S_{(t,1)} =: \\u{S}^{{1}}_t$, and sijections $\\varphi_t \\colon \\u{S}^{{0}}_t \\Rightarrow \\u{S}^{{1}}_t$. By the proposition, these sijections have a disjoint union that is a sijection $\\bigsqcup_{t \\in \\u T} \\u S^{{0}}_t \\Rightarrow \\bigsqcup_{t \\in \\u T} \\u S^{{1}}_t$.\n\n\\medskip\n\nBy the proposition, the relation\n$$\\u S \\approx \\u T \\iff \\mbox{there exists a sijection from } \\u S \\mbox{ to } \\u T$$\nis an equivalence {relation}.\n\n\\subsection*{Elementary signed sets and normal sijections}\n\nOften, we will be interested in disjoint unions of Cartesian products of signed intervals. An element of such a signed set is a pair, consisting of a tuple of integers and an element of the indexing signed set. Intuitively, the first one is ``more important'', as the second one serves just as an index. We formalize this notion in the following definition.\n\n\\begin{definition}\n A signed set $\\u A$ is \\emph{elementary of dimension $n$ and depth $0$} if its elements are in ${\\mathbb Z}^n$. A signed set $\\u A$ is \\emph{elementary of dimension $n$ and depth $d$}, $d \\geq 1$, if it is of the form\n $$\\bigsqcup_{t \\in \\u T} \\u S_t,$$\n where $\\u T$ is a signed set, and $\\u S_t$ are all signed sets of dimension $n$ and depth at most $d-1$, with the depth of at least one of them equal to $d-1$. A signed set $\\u A$ is \\emph{elementary of dimension $n$} if it is an elementary signed set of dimension $n$ and depth $d$ for some $d \\in {\\mathbb N}$.\\\\\n The \\emph{projection map} on an elementary set of dimension $n$ is the map\n $$\\xi \\colon \\u A \\to {\\mathbb Z}^n$$\n defined as follows. If the depth of $\\u A$ is $0$, then $\\xi$ is simply the inclusion map. Once $\\xi$ is defined on elementary signed sets of depth $< d$, and the depth of $\\u A$ is $d$, then $\\u A = \\bigsqcup_{t \\in \\u T} \\u S_t$, where $\\xi$ is defined on all $\\u S_t$. Then define $\\xi(s,t) = \\xi(s)$ for $(s,t) \\in \\u A$.\\\\\n A sijection $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$ between elementary signed sets $\\u T$ and $\\u{\\widetilde T}$ of the same dimension is \\emph{normal} if $\\xi(\\psi(t)) = \\xi(t)$ for all $t \\in \\u T \\sqcup \\u{\\widetilde T}$.\n\\end{definition}\n\nSimple examples of elementary signed sets are $\\si a c$, $\\si a b \\sqcup \\si {b+1} c$ and $\\si a c \\sqcup (\\si a b \\sqcup \\si {b+1} c)$. They are all of dimension $1$ and depth $0$, $1$ and $2$, respectively.\\footnote{{To avoid ambiguity, we should consider signed intervals in this case to be subsets of ${\\mathbb Z}^1$ ($1$-tuples of integers), not ${\\mathbb Z}$. Otherwise, $\\si{0}{1} \\sqcup \\si{2}{3} = (\\{(0,0),(1,0),(2,1),(3,1)\\},\\emptyset)$, and this can be seen either as an elementary set of dimension $1$ and depth $1$, or as an elementary signed set of dimension $2$ and depth $0$. So the interpretation depends on the ``representation'' of the set as disjoint union. Instead, we should understand $\\si{0}{1} \\sqcup \\si{2}{3}$ to mean $ (\\{((0),0),((1),0),((2),1),((3),1)\\},\\emptyset)$, with dimension $1$ and depth $1$. For coding, the distinction is important, but in the paper we nevertheless think of elements of signed intervals as integers.}} It is easy to see that the sijection $\\alpha_{a,b,c}$ from Problem \\ref{prob:alpha} is normal.\n\nLet us illustrate this with the example $a = 1$, $b = 5$, $c = 3$. We have $\\si a c = (\\{1,2,3\\},\\emptyset)$ and $\\si a b \\sqcup \\si {b+1} c = (\\{(1,0),(2,0),(3,0),(4,0),(5,0)\\},\\{(4,1),(5,1)\\})$. The sijection $\\alpha_{1,5,3}$ is the involution on $\\si 1 3 \\sqcup (\\si 1 5 \\sqcup \\si {6} 3)$ defined by\n$$(1,0) \\leftrightarrow ((1,0),1), \\quad (2,0) \\leftrightarrow ((2,0),1), \\quad (3,0) \\leftrightarrow ((3,0),1),$$\n$$((4,0),1) \\leftrightarrow ((4,1),1), \\quad ((5,0),1) \\leftrightarrow ((5,1),1).$$\nSince $\\xi(i,0) = i$ for $i = 1,2,3$, $\\xi((i,0),1) = i$ for $i = 1,2,3,4,5$ and $\\xi((i,1),1) = i$ for $i=4,5$, $\\alpha_{1,5,3}$ is indeed normal.\n\n\\medskip\n\nOther examples of elementary signed sets appear in the statements of Problems \\ref{prob:beta} and \\ref{prob:gamma} (in both cases, they are of dimension $n-1$).\n\n\\medskip\n\nNormality is preserved under Cartesian product, disjoint union etc. For example, the sijection\n\\begin{multline*}\n  \\si{a_1}{c_1} \\times \\si{a_2}{c_2} \\Rightarrow \\\\\n  \\si{a_1}{b_1} \\times \\si{a_2}{b_2} \\sqcup \\si{a_1}{b_1} \\times \\si{b_2+1}{c_2} \\sqcup \\si{b_1+1}{c_1} \\times \\si{a_2}{b_2} \\sqcup \\si{b_1+1}{c_1} \\times \\si{b_2+1}{c_2},\n\\end{multline*}\nobtained by using $\\alpha_{a_1,b_1,c_1} \\times \\alpha_{a_2,b_2,c_2}$ and distributivity on disjoint unions, is normal.\n\n\\medskip\n\nThe main reason normal sijections are important is that they give a very natural special case of Proposition \\ref{prop:sijections} (3). Suppose that $\\u T$ and $\\u{\\widetilde T}$ are elementary signed sets of dimension $n$, and that $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$ is a normal sijection. Furthermore, suppose that we have a signed set $\\u S_{\\q k}$ for every $\\q k \\in {\\mathbb Z}^n$. Then we have a sijection\n$$\\bigsqcup_{t \\in \\u T} \\u S_{\\xi(t)} \\Rightarrow \\bigsqcup_{t \\in \\u{\\widetilde T}} \\u S_{\\xi(t)}.$$\nIndeed, Proposition \\ref{prop:sijections} gives us a sijection provided that we have a sijection $\\varphi_t \\colon \\u S_{\\xi(t)} \\Rightarrow \\u S_{\\xi(\\psi(t))}$ satisfying $\\varphi_{\\psi(t)} = \\varphi_t^{-1}$ for every $t \\in \\u T \\sqcup \\u{\\widetilde T}$. But since $\\xi(\\psi(t)) = \\xi(t)$, we can take $\\varphi_t$ to be the identity.\n\n\\comment{The following observation is crucial: Suppose $\\u T \\approx \\u {\\widetilde T}$, then in general we do not have\n\\begin{equation}\n\\label{indexset_equivalence}\n\\bigsqcup_{t \\in \\u T} S_t \\approx \\bigsqcup_{t \\in \\u {\\widetilde T}} S_t.\n\\end{equation}\nHowever, if $\\u T=\\si{a}{c}$ and $\\u {\\widetilde T}=\\si{a}{b} \\sqcup \\si{b+1}{c}$ (see Problem~\\ref{prob:alpha}), then the identity is true because either $\\si{a}{b}$ and\n$\\si{b+1}{c}$ are disjoint (in which case the two boxes have the same sign) or one set is contained in the other (in which case the two boxes have opposite sign).}\n\n\\section{Some sijections on signed boxes} \\label{sec:sb}\n\nThe first sijection in this section will serve as the base of induction for Problem \\ref{prob:pi}.\n\n\\comment{\n\n\\begin{example} \\label{ex:toempty}\n  For $a,b \\in {\\mathbb Z}$, define $\\u T = \\si{a+1}{b+1} \\times \\si a b$. Then $\\psi: \\u T \\Rightarrow \\u T$, defined by $\\psi((l_1,l_2),i) = ((l_2+1,l_1-1),1-i)$ for $l_1 \\in \\si{a+1}{b+1}, l_2 \\in \\si a b, i \\in \\{0,1\\}$, is a (non-normal) sijection. Define $\\u S_{((l_1,l_2),i)} = (-1)^i \\si{l_1}{l_2}$. Since $\\u S_{((l_2+1,l_1-1),i)} = - \\u S_{((l_1,l_2),i)}$ and $\\u S_{((l_1,l_2),1-i)} = - \\u S_{((l_1,l_2),i)}$, we have $\\u S_{\\psi((l_1,l_2),i)} = \\u S_{((l_1,l_2),i)}$. Therefore we can use Proposition \\ref{prop:sijections} with $\\varphi_t = \\id$, and we obtain a sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} -\\si{l_1}{l_2},$$\n  which is equivalent to a sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\u \\emptyset.$$\n\\end{example}}\n\n\\begin{example} \\label{ex:toempty}\n  For $a,b \\in {\\mathbb Z}$, we have a normal sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\u \\emptyset$$\n  defined by $\\varphi((x,(l_1,l_2)),0) = ((x,(l_2+1,l_1-1)),0)$. It is well defined because $(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b$ if and only if $(l_2+1,l_1-1) \\in \\si{a+1}{b+1} \\times \\si a b$, and because $x \\in \\si{l_1}{l_2}$ if and only if $x \\in \\si{l_2+1}{l_1-1}$.\n\\end{example}\n\nNote that the $0$ as the second coordinate in the example comes from the fact that a sijection in question is an involution on the disjoint union\n$$\\left(\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2}\\right) \\sqcup \\u \\emptyset = \\left(\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2}\\right) \\times \\u{\\{0\\}} \\cup \\u \\emptyset \\times \\u{\\{1\\}}.$$\nWe could be a little less precise and write $\\varphi(x,(l_1,l_2)) = (x,(l_2+1,l_1-1))$ without causing confusion.\n\n\\medskip\n\nThe following generalizes the construction of Problem \\ref{prob:alpha}; indeed, for $n = 2$ the construction gives a sijection from $[a_1,b_1]$ to $[a_1,x] \\sqcup (-[b_1+1,x])$.\n\n\\begin{problem} \\label{prob:beta}\n Given $\\q a =(a_1,\\ldots,a_{n-1}) \\in {\\mathbb Z}^{n-1}$, $\\q b=(b_1,\\ldots,b_{n-1}) \\in {\\mathbb Z}^{n-1}$, $x \\in {\\mathbb Z}$, construct a normal sijection\n $$ \\beta = \\beta_{\\q a,\\q b,x} \\colon \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x,$$\n where $\\u S_i = (\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$\n\\end{problem}\n\n{Note that $(\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$ can be identified with\n$(\\{a_i\\},\\{b_i+1\\})$ if $a_i \\not= b_i+1$.}\n\n\\begin{proof}[Construction]\n The proof is by induction, with the case $n = 1$ being trivial and the case $n=2$ was constructed in Problem~\\ref{prob:alpha}. Now, for $n \\ge 3$,\n \\begin{multline*}\n   \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\approx \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_2,\\ldots,l_{n-1}) \\in \\u S_2 \\times \\cdots \\times \\u S_{n-1}} \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}}x \\\\\n   \\approx \\left( \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} \\si{a_2}{l_3} \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\ \\sqcup \\left( \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} (-\\si{b_2+1}{l_3}) \\times \\cdots \\times \\si{l_{n-1}}x \\right),\n\\end{multline*}\nwhere we used induction for the first equivalence, and distributivity and the fact that $S_2 = (\\{a_2\\},\\emptyset) \\sqcup (\\emptyset,\\{b_2+1\\})$ for the second equivalence. By Problem~\\ref{prob:alpha} and Proposition~\\ref{prop:sijections} (2), there exists a sijection from the last expression to\n\\begin{multline*}\n   \\left( \\left(\\si{a_1}{a_2} \\sqcup (-\\si{b_1+1}{a_2}) \\right) \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} \\si{a_2}{l_3} \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\\n   \\sqcup \\left( \\left( \\si{a_1}{b_2+1} \\sqcup (-\\si{b_1+1}{b_2+1}) \\right) \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} (-\\si{b_2+1}{l_3}) \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\\n   \\approx \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\cdots \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x,\n\\end{multline*}\nwhere for the last equivalence we have again used distributivity. Normality follows from the normality of all the sijections involved in the construction.\n\\end{proof}\n\n\\begin{problem} \\label{prob:gamma}\n Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a normal sijection\n \\begin{multline*}\n   \\gamma = \\gamma_{\\q k,x} \\colon \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n   \\Rightarrow \\bigsqcup_{i = 1}^n \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n   \\sqcup \\bigsqcup_{i = 1}^{n-2} \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots .\n   \\end{multline*}\n \\end{problem}\n\n\\begin{proof}[Construction] The proof is by induction with respect to $n$. The case $n=1$ is trivial, and $n=2$ is Problem~\\ref{prob:alpha}. Now take $n > 2$. By the induction hypothesis (for $(k_1,\\ldots,k_{n-1})$ and $x+1$), we have\n\\begin{multline*}\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\approx \\bigg( \\bigsqcup_{i = 1}^{n-1} \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-2}}{k_{n-1}}\\\\\n  \\sqcup \\bigsqcup_{i = 1}^{n-3} \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\cdots \\times \\si{k_{n-2}}{k_{n-1}} \\bigg) \\times \\si{k_{n-1}}{k_n}.\n  \\end{multline*}\nWe use distributivity. We keep all terms except the one corresponding to $i = n-1$ in the first part. Because\n\\begin{multline*}\n\\si{k_{n-2}}{x+1} \\times \\si{k_{n-1}}{k_n} \\approx \\si{k_{n-2}}{x+1} \\times (\\si{k_{n-1}}x \\sqcup \\si{x+1}{k_n}) \\\\\n\\approx (\\si{k_{n-2}}{k_{n-1}} \\sqcup \\si{k_{n-1}+1}{x+1}) \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-2}}{x+1} \\times \\si{x+1}{k_n} \\\\\n{\\approx \\si{k_{n-2}}{k_{n-1}} \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-1}+1}{x+1} \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-2}}{x+1} \\times \\si{x+1}{k_n}},\n\\end{multline*}\nwe obtain the required Cartesian products for the first term on the right-hand side at $i = n$, the second term at $i = n-2$, and the first term at $i = n-1$. Again, normality follows from the fact that $\\alpha$ is normal.\n\\end{proof}\n\n\n\\section{Gelfand-Tsetlin patterns} \\label{sec:gt}\n\nUsing our definition of a disjoint union of {signed sets}, it is easy to define generalized Gelfand-Tsetlin patterns, or GT patterns for short (compare with \\cite{Fis05}).\n\n\\begin{definition}\n For $k \\in {\\mathbb Z}$, define ${\\GT(k) = (\\{\\cdot\\},\\emptyset)}$, \n and for $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$, define {recursively}\n $$\\GT(\\q k) = \\GT(k_1,\\ldots,k_n) = \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(l_1,\\ldots,l_{n-1}).$$\n\\end{definition}\n\nIn particular, $\\GT(a,b) \\approx \\si a b$.\n\n\\medskip\n\nOf course, one can think of an element of $\\GT(\\q k)$ in the usual way, as a triangular array $A=(A_{i,j})_{1 \\leq j \\leq i \\le n}$ of $\\binom {n+1}2$ numbers,\narranged as\n$$\n\\begin{array}{ccccccccc}\n &&&& A_{1,1} &&&& \\\\\n &&& A_{2,1} && A_{2,2} &&& \\\\\n && A_{3,1} && A_{3,2} && A_{3,3} && \\\\\n & \\iddots & \\vdots & \\ddots & \\vdots & \\iddots & \\vdots & \\ddots &  \\\\\nA_{n,1} && A_{n,2} && \\ldots && \\ldots && A_{n,n},\n \\end{array}\n$$\nso that $A_{i+1,j} \\leq A_{i,j} \\leq A_{i+1,j+1}$ or $A_{i+1,j} > A_{i,j} > A_{i+1,j+1}$ for $1 \\leq j \\leq i < n$, and $A_{n,i}=k_i$. The sign of such an array is $(-1)^m$, where $m$ is the number of $(i,j)$ with $a_{i,j} > a_{i,j+1}$.\n\n\\medskip\n\nSome crucial sijections for GT patterns are given by the following constructions.\n\n\\begin{problem} \\label{prob:rho}\n  Given $\\q a =(a_1,\\ldots,a_{n-1}) \\in {\\mathbb Z}^{n-1}$, $\\q b=(b_1,\\ldots,b_{n-1}) \\in {\\mathbb Z}^{n-1}$, $x \\in {\\mathbb Z}$, construct a sijection\n  $$ \\rho = \\rho_{\\q a,\\q b,x} \\colon \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(l_1,\\ldots,l_{n-1},x),$$\n  where $\\u S_i = (\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$.\n\\end{problem}\n\\begin{proof}[Construction]\n In Problem \\ref{prob:beta}, we constructed a normal sijection\n $$ \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x.$$\n By Proposition~\\ref{prop:sijections} (3) (see the comment at the end of Section \\ref{sec:ss}), this gives a sijection\n $$ \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{\\q m \\in \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x} \\GT(\\q m).$$\n By basic sijection constructions, we get that this is equivalent to\n $$\\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\bigsqcup_{\\q m \\in \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x} \\GT(\\q m),$$\n and by definition of $\\GT$, this is equal to $\\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(l_1,\\ldots,l_{n-1},x)$.\n\\end{proof}\n\nThe result is important because while it adds a dimension to GT patterns, it (typically) greatly reduces the size of the indexing signed set. {In fact, there is an analogy to the fundamental theorem of calculus: instead of extending the disjoint union over the entire signed box, it suffices to consider the boundary; $x$ corresponds in a sense to the constant of integration.}\n\n\\begin{problem} \\label{prob:pi}\n  Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $i$, $1 \\leq i \\leq n-1$, construct a sijection\n $$\\pi = \\pi_{\\q k,i} \\colon \\GT(k_1,\\ldots,k_n) \\Rightarrow -\\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,k_i-1,k_{i+2},\\ldots,k_n).$$\n Given $\\q a =(a_1,\\ldots,a_n) \\in {\\mathbb Z}^n$,  $\\q b =(b_1,\\ldots,b_n) \\in {\\mathbb Z}^n$ such that for some $i$, $1 \\leq i \\leq n-1$, we have $a_{i+1} = a_i - 1$ and $b_{i+1} = b_i - 1$, construct a sijection\n $$\\sigma = \\sigma_{\\q a,\\q b,i} \\colon \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_n}{b_n}} \\GT(\\q l) \\Rightarrow \\u \\emptyset.$$\n\\end{problem}\n\\begin{proof}[Construction]\n The proof is by induction, with the induction step for $\\pi$ using $\\sigma$ and vice versa. For $n = 1$, there is nothing to prove. For $n=2$ and $i=1$, the existence of $\\pi$ follows from the statement $\\si{k_1}{k_2} = -\\si{k_2+1}{k_1-1}$, and $\\sigma$ was constructed in Example \\ref{ex:toempty}. Assume that $n > 2$ and $1 < i < n-1$. We have\n$$\\GT(k_1,\\ldots,k_n) =  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}).$$\nBy using $\\id \\times \\cdots \\times \\id \\times \\alpha_{k_{i-1},k_{i+1}+1,k_i} \\times \\id \\times \\alpha_{k_{i+1},k_i-2,k_{i+2}} \\times \\id \\times \\cdots \\times \\id$ and distributivity, we get a normal sijection\n\\begin{multline*}\n\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\Rightarrow \\\\\n\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}\n\\end{multline*}\n By Proposition~\\ref{prop:sijections} (3), this gives a sijection\n \\begin{multline*}\n  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}) \\Rightarrow \\\\\n   \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} - \\GT(l_1,\\ldots,l_{n-1}) \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1})  \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1})\n \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}).\n \\end{multline*}\n By definition, the first signed set on the right-hand side is $-\\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,k_i-1,k_{i+2},\\ldots,k_n)$. The other three disjoint unions all satisfy the condition needed for the existence of $\\sigma$ {(for $i$, for $i-1$ and for both, $i-1$ and $i$, respectively)}, and hence we can siject them to $\\u \\emptyset$.\\\\\n If $i = 1$ or $i = n-1$, the proof is similar but easier (as we only have to use $\\alpha$ once, and we get only two factors after using distributivity). Details are left to the reader.\\\\\n Now take $\\q l = (l_1,\\ldots,l_n)$ and $\\q{l'}=(l_1,\\ldots,l_{i-1},l_{i+1}+1,l_i-1,l_{i+2},\\ldots,l_n)$. The sijection $\\sigma$ can then be defined as\n $$\\sigma_{\\q a,\\q b,i}(A,\\q l) = \\begin{cases} (\\pi_{\\q l,i}(A),\\q l) & \\mbox{if }\\pi_{\\q l,i}(A) \\in \\GT(\\q l) \\\\ (\\pi_{\\q l,i}(A),\\q {l'}) & \\mbox{if } \\pi_{\\q {l},i}(A) \\in \\GT(\\q {l'})  \\end{cases}.$$\n It is easy to check that this is a sijection. Compare with Example \\ref{ex:toempty}.\n \\comment{These follow immediately from such sijections on each of the following two signed sets,\n $$\n \\bigsqcup_{(l_{i-1},l_i) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}}  \\GT(l_1,\\ldots,l_{n-1})\n$$\nand\n$$\n \\bigsqcup_{ (l_i,l_{i+1}) \\in  \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2}}  \\GT(l_1,\\ldots,l_{n-1}),\n$$\nfixing arbitrary integers $l_1,\\ldots,l_{i-2},l_{i+1},\\ldots,l_{n-1}$ in the first case and $l_1,\\ldots,l_{i-1},l_{i+2},\\ldots,l_{n-1}$ in the second case. As for the first case, we let\n$$\nA \\times \\u {\\{ (l_{i-1},l_{i}) \\}}  \\in  \\bigsqcup_{(l_{i-1},l_i) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}}  \\GT(l_1,\\ldots,l_{n-1})\n$$\nand map it to\n$$\n\\begin{cases}\n\\,\\, \\pi_{\\q l,i-1} (A) \\times \\u {\\{ (l_{i-1},l_{i}) \\}} & \\mbox{if } \\pi_{\\q l,i-1}(A) \\in  \\GT(l_1,\\ldots,l_{n-1}) \\\\\u00a0\n-\\pi_{\\q l,i-1} (A) \\times \\u {\\{ (l_{i}+1,l_{i-1}-1) \\}} & \\mbox{if } \\pi_{\\q l,i-1}(A) \\in  -\\GT(l_1,\\ldots,l_{i-2},l_i+1,l_{i-1}-1,l_{i+1},\\ldots,l_{n-1})\n\\end{cases}\n$$\nwhich is feasible since $(l_{i}+1,l_{i-1}-1) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}$, and $\\u {\\{ (l_{i-1},l_{i}) \\}}$ and\n$\\u {\\{ (l_{i}+1,l_{i-1}-1) \\}}$ have the same sign in $\\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}$. The second case is}\n\\end{proof}\n\n\\begin{problem} \\label{prob:tau}\n Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n $$\\tau = \\tau_{\\q k,x} \\colon \\GT(k_1,\\ldots,k_n) \\Rightarrow \\bigsqcup_{i = 1}^n \\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n).$$\n\\end{problem}\n\\begin{proof}[Construction]\n In Problem \\ref{prob:gamma}, we constructed a normal sijection\n \\begin{multline*}\n   \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\Rightarrow \\bigsqcup_{i = 1}^n \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}\\\\\n   \\sqcup \\bigsqcup_{i = 1}^{n-2} \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots ,\n   \\end{multline*}\nwhich gives a sijection\n\\begin{multline*}\\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{i = 1}^n \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}}\\GT(\\q l) \\\\\n\\sqcup \\bigsqcup_{i = 1}^{n-2} \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l).\n\\end{multline*}\nAll disjoint unions in the second term satisfy the conditions for the existence of $\\sigma$ from Problem \\ref{prob:pi}, so we can siject them to $\\u \\emptyset$. This gives a sijection\n $$\\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{i = 1}^n \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}}\\GT(\\q l),$$\n which is, by the definition of $\\GT$, a sijection $\\GT(k_1,\\ldots,k_n) \\Rightarrow \\bigsqcup_{i = 1}^n \\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n)$.\n \\comment{Finally, for each $i$, apply the sijection\n $$\\pi_{(k_1,\\ldots,k_{i-1},k_{i+1}+1,\\ldots,k_{n-1}+1,x+1,k_n),n-1} \\circ \\cdots \\circ \\pi_{(k_1,\\ldots,k_{i-1},k_{i+1}+1,x+n-i-1,k_{i+2},\\ldots,k_n),i+1} \\circ \\pi_{(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n),i}$$\n from $\\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n)$ to $(-1)^{n-i} \\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,\\ldots,k_n+1,x)$.}\n\\end{proof}\n\n\n\\section{Combinatorics of the monotone triangle recursion} \\label{sec:recursion}\n\n\\subsection*{Monotone triangles}\n\nSuppose that $\\q k = (k_1,\\ldots,k_n)$ and $\\q l = (l_1,\\ldots,l_{n-1})$ are two sequences of integers. We say that $\\q l$ \\emph{interlaces} $\\q k$, $\\q l \\prec \\q k$, if the following holds:\n\\begin{enumerate}\n  \\item for every $i$, $1 \\leq i \\leq n-1$, $l_i$ is in the closed interval between $k_i$ and $k_{i+1}$;\n  \\item if $k_{i-1} \\leq k_i \\leq k_{i+1}$ for some $i$, $2 \\leq i \\leq n-1$, then $l_{i-1}$ and $l_i$ cannot both be $k_i$;\n  \\item if $k_i > l_i = k_{i+1}$, then $i \\leq n-2$ and $l_{i+1} = l_i = k_{i+1}$;\n  \\item if $k_i = l_i > k_{i+1}$, then $i \\geq 2$ and $l_{i-1} = l_i = k_i$.\n\\end{enumerate}\n\nFor example, if $k_1 < k_2 < \\ldots < k_n$, then $l_i \\in [k_i,k_{i+1}]$ and $l_1 < l_2 < \\ldots < l_{n-1}$.\n\n\\medskip\n\nA \\emph{monotone triangle of size $n$} is a map $T \\colon \\{(i,j) \\colon 1 \\leq j \\leq i \\leq n \\} \\to {\\mathbb Z}$ so that line $i-1$ (i.e.~the sequence $T_{i-1,1},\\ldots,T_{i-1,i-1}$) interlaces line $i$ (i.e.~the sequence $T_{i,1},\\ldots,T_{i,i}$).\n\\begin{example} {The following is a monotone triangles of size $5$:}\n$$\\begin{array}{ccccccccc}\n&&&& 4 &&&& \\\\\n&&& 3 && 5 &&& \\\\\n&& 3 && 4 && 5 && \\\\\n& 3 & & 3 & & 4 & & 5 & \\\\\n5 & & 3 & & 1 & & 4 & & 6\n\\end{array}\n$$\n\\end{example}\n{This notion of (generalized) monotone triangle was introduced in \\cite{Rie13}. Other notions appeared in \\cite{Fis12}.}\u00a0\n\n\n\\medskip\n\nThe \\emph{sign} of a monotone triangle $T$ is $(-1)^r$, where $r$ is the sum of:\n\\begin{itemize}\n  \\item the number of strict descents in the rows of $T$, i.e.~the number of pairs $(i,j)$ so that $1 \\leq j < i \\leq n$ and $T_{i,j} > T_{i,j+1}$, and\n  \\item the number of $(i,j)$ so that $1 \\leq j \\leq i - 2$, $i \\leq n$ and $T_{i,j} > T_{i-1,j} = T_{i,j+1} = T_{i-1,j+1} > T_{i,j+2}$.\n\\end{itemize}\n{The sign of our example is $-1$.}\n\n\\medskip\n\nWe denote the signed set of all monotone triangles with bottom row $\\q k$ by ${\\MT(\\q k)}$.\n\n\\medskip\n\nIt turns out that $\\MT(\\q k)$ satisfies a recursive ``identity''. Let us define the signed set of \\emph{arrow rows of order $n$} as\n$$\\AR_n = (\\{\\nearrow,\\nwarrow\\},\\{\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\})^{n}.$$\nAlternatively, we can think of them as rows of length $n$ with elements $\\nwarrow, \\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$, where the positive elements are precisely those with an even number of $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$'s.\n\n\\medskip\n\nThe role of an arrow row $\\mu$ of order $n$ is that it induces a deformation of $\\si{k_1}{k_2} \\times \\si{k_2}{k_3} \\times \\cdots \\times \\si{k_{n-1}}{k_n}$ as follows. Consider\n$$\n\\begin{array}{ccccccccccc}\n & \\si{k_1}{k_2} &   & \\si{k_2}{k_3} &  &  \\ldots &  & \\si{k_{n-2}}{k_n-1} &  & \\si{k_{n-1}}{k_n} & \\\\\n \\mu_1 & & \\mu_2  & & \\mu_3 &  &  \\ldots &   & \\mu_{n-1} & & \\mu_{n} ,\n \\end{array}\n $$\nand if $\\mu_i \\in \\{\\nwarrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$ (that is we have an arrow pointing towards $\\si{k_{i-1}}{k_i}$) then\n$k_i$ is decreased by $1$ in $\\si{k_{i-1}}{k_i}$, while there is no change for this $k_i$ if $\\mu_i=\\nearrow$. If $\\mu_i \\in \\{\\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$ (that is we have an arrow pointing towards $[k_i,k_{i+1}]$) then $k_i$ is increased by $1$ in $[k_i,k_{i+1}]$, while there is no change for this $k_i$ if $\\mu_i=\\nwarrow$.\n\n\\medskip\n\nFor a more formal description, we let $\\delta_{\\nwarrow}(\\nwarrow) = \\delta_{\\nwarrow}(\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow) = \\delta_{\\nearrow}(\\nearrow) = \\delta_{\\nearrow}(\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow) = 1$ and $\\delta_{\\nwarrow}(\\nearrow) = \\delta_{\\nearrow}(\\nwarrow) = 0$, and we define\n$$e(\\q k,\\mu)  = \\si{k_1+\\delta_{\\nearrow}(\\mu_1)}{k_2-\\delta_{\\nwarrow}(\\mu_2)} \\times \\ldots \\times \\si{k_{n-1}+\\delta_{\\nearrow}(\\mu_{n-1})}{k_n-\\delta_{\\nwarrow}(\\mu_n)}.$$\nfor $\\q k = (k_1,\\ldots,k_n)$ and $\\mu \\in \\AR_n$.\n\n\\begin{problem} \\label{prob:Xi}\n  Given $\\q k = (k_1,\\ldots,k_n)$, construct a sijection\n  $$\\Xi = \\Xi_{\\q k} \\colon \\MT(\\q k) \\Rightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\MT(\\q l).$$\n\\end{problem}\n\\begin{proof}[Construction]\n All elements on the left are mapped to the right with $\\Xi$, while there are quite a few cancellations on the right-hand side. More specifically, take a monotone triangle $T$ with bottom row $\\q k$. Then $\\Xi(T) = ((T',\\q l),\\mu)$, where $T'$ is the monotone triangle we obtain from $T$ by deleting the last row, $\\q l$ is the bottom row of $T'$, and $\\mu = (\\mu_1,\\ldots,\\mu_n)$ is the arrow row defined as follows:\n \\begin{itemize}\n  \\item $\\mu_1 = \\nwarrow$;\n  \\item $\\mu_n = \\nearrow$;\n  \\item for $ 1 < i < n$, $\\mu_i$ is determined as follows:\n  \\begin{enumerate} \\item if $k_{i-1} \\leq l_{i-1} = k_i$, take $\\mu_i = \\nearrow$;  \\item if $k_{i-1} > l_{i-1} = k_i = l_{i} > k_{i+1}$, take $\\mu_i = \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$; \\item otherwise, take $\\mu_i = \\nwarrow$. \\end{enumerate}\n \\end{itemize}\n It is easy to check that $\\q l$ is indeed in $e(\\q k,\\mu)$.\n Note that in (1) and (2) of the third bullet point, $\\mu_i$ is forced if we require $\\q l \\in e(\\q k,\\mu)$. In (3), $\\mu_i=\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ would also be possible if and only if  $\\mu_i=\\nearrow$ would also be possible.\\\\\n On the other hand, for $((T',\\q l),\\mu)$, define $\\Xi((T',\\q l),\\mu)$ as follows. For the construction it is useful to keep in mind that $\\q l \\in e(\\q k, \\mu)$ implies that conditions (1) and (2) for $\\q l \\prec \\q k$ are satisfied.\n \\begin{itemize}\n  \\item if $\\mu_1 \\neq \\nwarrow$, take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $1$ by $\\nearrow$ and vice versa;\n  \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n \\neq \\nearrow$, take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $n$ by $\\nwarrow$ and vice versa;\n  \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n = \\nearrow$, and $\\q l \\not\\prec \\q k$, find the smallest $i$ between $2$ and $n-1$ such that:\n \\begin{itemize}\n   \\item condition (3) of $\\q l \\prec \\q k$ is not satisfied at $i$, i.e.\n   $k_{i-1} > l_{i-1} = k_{i} \\not= l_{i}$ (which implies $\\mu_i \\in \\{\\nwarrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$), or\n   \\item condition (4) of $\\q l \\prec \\q k$ is not satisfied at $i$, i.e.\n   $l_{i-1} \\not= k_i = l_i > k_{i+1}$ (which implies  $\\mu_i \\in \\{\\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$).\n\\end{itemize}\nThen take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nwarrow$ and vice versa in the first case, and replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nearrow$ and vice versa in the second case;\n \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n = \\nearrow$, and $\\q l \\prec \\q k$, find the smallest $i$ for an instance of (3) of the third bullet point in the first paragraph of the proof with $\\mu_i \\not= \\nwarrow$ (if such an $i$ exists). Then take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nwarrow$ and vice versa.\n \\end{itemize}\nIf no such $i$ exists, we take $\\Xi((T',\\q l),\\mu) = T$, where we obtain $T$ from $T'$ by adding $\\q k$ as the last row. It is easy to see that this is a well-defined sijection.\n\\end{proof}\n\n\n\\begin{remark} The previous construction could have been avoided by using alternative extensions of monotone triangles provided in \\cite{Fis12}. However, the advantage of the definition used in this paper is that it is more reduced than the others in the sense that it can obtained from these by cancelling elements using certain sign-reversing involutions.\n\\end{remark}\n\n\\subsection*{Arrow patterns and shifted GT patterns}\n\nDefine the signed set of \\emph{arrow patterns of order $n$} as\n$$\\AP_n = (\\{\\swarrow, \\searrow\\},\\{\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\})^{\\binom n 2}.$$\n\n\\medskip\n\nAlternatively, we can think of an arrow pattern of order $n$ as a triangular array $T=(t_{p,q})_{1 \\le p < q \\le n}$ arranged as\n$$T = \\begin{smallmatrix} & & & & t_{1,n} & & & & \\\\ & & & t_{1,n-1} & & t_{2,n} & & & \\\\ & & t_{1,n-2} & & t_{2,n-1}& & t_{3,n} & & \\\\ & \\vstretch{0.35}{\\udots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\ddots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\udots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\ddots} & \\\\ t_{1,2} & & t_{2,3}  & & \\ldots  & & \\ldots  & & t_{n-1,n}  \\end{smallmatrix},$$\nwith $t_{p,q} \\in \\{\\swarrow, \\searrow, \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\}$, and the sign of an arrow pattern is $1$ if the number of $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$'s is even and $-1$ otherwise.\n\n\\medskip\n\nThe role of an arrow pattern of order $n$ is that it induces a deformation of $(k_1,\\ldots,k_n)$, which can be thought of as follows. Add $k_1,\\ldots,k_n$ as bottom row of $T$ (i.e., $t_{i,i}=k_i$), and for each $\\swarrow$ or$ \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ which is in the same $\\swarrow$-diagonal as $k_i$ add $1$ to $k_i$, while for each $\\searrow$ or $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ which is in the same $\\searrow$-diagonal as $k_i$ subtract $1$ from $k_i$.\nMore formally, letting $\\delta_{\\swarrow}(\\swarrow) = \\delta_{\\swarrow}(\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow) = \\delta_{\\searrow}(\\searrow) = \\delta_{\\searrow}(\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow) = 1$ and $\\delta_{\\swarrow}(\\searrow) = \\delta_{\\searrow}(\\swarrow) = 0$, we set\n$$c_i(T) = \\sum_{j=i+1}^{n} \\delta_{\\swarrow}(t_{i,j}) - \\sum_{j=1}^{i-1} \\delta_{\\searrow}(t_{j,i}) \\mbox{ and } d(\\q k,T) = (k_1+c_1(T),k_2+c_2(T),\\ldots,k_n+c_n(T))$$\nfor $\\q k = (k_1,\\ldots,k_n)$ and $T \\in \\AP_n$.\n\n\\medskip\n\nFor $\\q k =(k_1,\\ldots,k_n)$ define \\emph{shifted Gelfand-Tsetlin patterns}, or SGT patterns for short, as the following disjoint union of GT patterns over arrow patterns of order $n$:\n$$\n\\SGT(\\q k) = \\bigsqcup_{T \\in \\AP_n} \\GT(d(\\q k,T))\n$$\n\n\n\\medskip\n\nConsidering that $|(\\{\\swarrow, \\searrow\\},\\{\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\})| = 1$ and therefore $|\\AP_n| = 1$, the following is not surprising.\n\n\\begin{problem} \\label{prob:Psi}\n Given $n$ and $i$, $1 \\leq i \\leq n$, construct a sijection\n $$\\Psi = \\Psi_{n,i} \\colon  \\AP_{n-1} \\Rightarrow \\AP_n.$$\n\\end{problem}\n\\begin{proof}[Construction]\n For $T \\in \\AP_{n-1}$, take $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$ to be the arrow pattern defined by\n $$t'_{p,q} = \\begin{cases} t_{p,q} & \\mbox{if } p < q < i \\\\ t_{p,q-1} & \\mbox{if } p < i < q \\\\ t_{p-1,q-1} & \\mbox{if } i < p < q \\\\ \\searrow & \\mbox{if } p < q = i \\\\ \\swarrow & \\mbox{if } i = p < q \\end{cases}.$$\n{An example for $n=6$ and $i=4$ is\n$$\n\\begin{array}{ccccccc}\u00a0\n&&& \\searrow &&& \\\\\n&& \\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\\\\n& \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\swarrow && \\swarrow & \\\\\n\\searrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\searrow && \\swarrow\n\\end{array} \\quad \\stackrel{\\Psi}{\\Rightarrow} \\quad\n\\begin{array}{ccccccccc}\u00a0\n&&&& \\searrow &&&& \\\\\n&&& \\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow &&& \\\\\n&& {\\color{red} \\searrow} && \\swarrow &&  \\swarrow && \\\\\n& \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow &&  {\\color{red} \\searrow} && \\searrow && {\\color{red} \\swarrow}  & \\\\\n\\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && {\\color{red} \\searrow}\u00a0&& {\\color{red} \\swarrow} && \\swarrow\n\\end{array},\n$$\nwhere the new arrows are indicated in red.}\n If $T \\in \\AP_{n}$, $t_{p,i} = \\searrow$ for $p = 1,\\ldots,i-1$, $t_{i,q} = \\swarrow$ for $q = i+1,\\ldots,n$, take $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n-1}$, where\n $$t'_{p,q} = \\begin{cases} t_{p,q} & \\mbox{if } p < q < i \\\\ t_{p,q+1} & \\mbox{if } p < i \\leq q \\\\ t_{p+1,q+1} & \\mbox{if } i \\leq p < q \\end{cases}.$$\nOtherwise, there either exists $p$ so that $t_{p,i} \\neq \\searrow$, or there exists $q$ so that $t_{i,q} \\neq \\swarrow$. In the first case, define $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$, where $t'_{p,i}=\\swarrow$ if $t_{p,i} = \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ and $t'_{p,i}=\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if $t_{p,i} = \\swarrow$, and all other array elements are equal. In the second case, define $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$, where $t'_{i,q}=\\searrow$ if $t_{i,q} = \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ and $t'_{i,q}=\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if $t_{i,q} = \\searrow$, and all other array elements are equal. It is easy to see that this is a sijection.\n\\end{proof}\n\nThe difficult part of this paper is to prove that $\\SGT$ satisfies the same ``recursion'' as $\\MT$. While the proof of the recursion was easy for monotone triangles, it is very involved for shifted GT patterns, and needs almost {all} the sijections we have constructed in this and previous sections.\n\n\\begin{problem}\nGiven $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n$$\\Phi = \\Phi_{\\q k,x} \\colon \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\Rightarrow \\SGT(\\q k).$$\n\\end{problem}\n\\begin{proof}[Construction]\nTo make the construction of $\\Phi$ a little easier, we will define it as the composition of several sijections. The first one will reduce the indexing sets (from a signed box to its ``corners'') using Problem \\ref{prob:rho}. The second one increases the order of the arrow patterns using the sijection from Problem \\ref{prob:Psi}. The third one further reduces the indexing set (from a signed set with $2^{n-1}$ elements to $\\si{1}{n}$). The last one gets rid of the arrow row and then uses Problem \\ref{prob:tau}.\\\\\nFor $\\mu \\in \\AR_n$, define $\\u S_i = (\\{k_i+\\delta_{\\nearrow}(\\mu_i)\\},\\emptyset) \\sqcup (\\emptyset,\\{k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1\\})$. Then $\\Phi$ is the composition of sijections\n\\begin{multline*}\n  \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\\\\\stackrel{\\Phi_1}\\Longrightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\\\\n\\stackrel{\\Phi_2}\\Longrightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{\\q m \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\\\ \\stackrel{\\Phi_3} \\Longrightarrow\n\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{i=1}^n \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots) \\\\\n\\stackrel{\\Phi_4}\\Longrightarrow \\SGT(\\q k), \\hspace*{12cm}\n\\end{multline*}\nwhere $\\Phi_1$, $\\Phi_2$, $\\Phi_3$, and $\\Phi_4$ are constructed as follows.\\\\\n\\emph{Construction of $\\Phi_1$.} By definition of $\\SGT$, we have\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) = \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\bigsqcup_{T \\in \\AP_{n-1}} \\GT(d(\\q l,T)).$$\nBy switching the inner disjoint unions, we get a sijection to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)}  \\GT(d(\\q l,T)).$$\nThere is an obvious sijection from this signed set to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q l \\in d(e(\\q k,\\mu),T)}  \\GT(\\q l),$$\n{by abuse of notation setting}\n$${ d(\\si{x_1}{y_1} \\times \\ldots \\si{x_{n-1}}{y_{n-1}},T)=\\si{x_1+c_1(T)}{y_1+c_1(T)} \\times \\cdots \\times \\si{x_{n-1}+c_{n-1}(T)}{y_{n-1}+c_{n-1}(T)}.}$$\nNow for each $\\mu$ and $T$, use the map $\\rho$ from Problem \\ref{prob:rho} for $a_i = k_i+\\delta_{\\nearrow}(\\mu_i)+c_i(T)$, $b_i = k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_i(T)$, and $x$. We get a sijection to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u {S}'_1 \\times \\cdots \\u{S}'_{n-1}} \\GT(m_1,\\ldots,m_{n-1},x),$$\nwhere $\\u{S}'_i = (\\{k_i+\\delta_{\\nearrow}(\\mu_i)+c_i(T)\\},\\emptyset) \\sqcup (\\emptyset,\\{k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_i(T)+1\\})$. Finally, there is an obvious sijection from this signed set to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u {S}_1 \\times \\cdots \\u{S}_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x).$$\n\\emph{Construction of $\\Phi_2$.} In Problem \\ref{prob:Psi}, we constructed sijections $\\Psi_{n,i} \\colon \\AP_{n-1} \\Rightarrow \\AP_n$. We construct $\\Phi_2$ by using Proposition \\ref{prop:sijections}~ (3) for $\\psi = \\Psi_{n,n}$, $\\u T = \\AP_{n-1}$, $\\u{\\widetilde T} = \\AP_n$,\n$$\\u S_T =  \\bigsqcup_{\\q m \\in \\u {S}_1 \\times \\cdots \\u{S}_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\quad \\mbox{for } T \\in \\AP_{n-1} \\sqcup \\AP_n$$\nand $\\varphi_T = \\u\\id$. This is well defined because $c_i(T) = c_i(\\Psi_{n,n}(T))$ for $T \\in \\AP_{n-1} \\sqcup \\AP_n$ and $i=1,\\ldots,n-1$.\\\\%Since $x(\\Phi(T)) = x(T)$ for all $T \\in \\AP_{n-1} \\sqcup \\AP_n$, this is well defined. \\\\\n\\emph{Construction of $\\Phi_3$.} Let $\\eta$ be the involution that maps $\\swarrow \\leftrightarrow \\nearrow, \\searrow \\leftrightarrow \\nwarrow, \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\leftrightarrow \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$. The elements of the signed set $\\u S = \\u {S}_1 \\times \\cdots  \\times \\u{S}_{n-1}$ are $(n-1)$-tuples of elements that are either $(k_i+\\delta_{\\nearrow}(\\mu_i),0)$ or $(k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1,1)$. Define $\\u S'$ as the subset of $\\u S$ containing tuples of the form $(\\ldots,(m_i,1),(m_{i+1},0),\\ldots)$, i.e.~the ones where we choose $k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1$ in position $i$ and $k_{i+1}+\\delta_{\\nearrow}(\\mu_{i+1})$ in position $i+1$ for some $i$. Then we can define a sijection\n$$ \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{\\q m \\in \\u S'} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\Rightarrow \\u \\emptyset$$\nas follows: given $\\mu \\in \\AR_n$, $T \\in \\AP_n$, $\\q m = (\\ldots,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1,k_{i+1}+\\delta_{\\nearrow}(\\mu_{i+1}),\\ldots)$ (and $i$ is the smallest index where this happens), $A \\in \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x)$, map $(((A,\\q m),T),\\mu)$ to $(((A',\\q m'),T'),\\mu')$, where:\n\\begin{itemize}\n \\item $A' = \\pi_{i,n}(A)$;\n \\item $T'$ is $T$ if $A'$ has the same bottom row as $A$; otherwise, $T'$ is obtained from $T$ by interchanging {$t_{i,j}$ and $t_{i+1,j}$ for $j>i+1$ as well as $t_{j,i}$ and $t_{j,i+1}$ for $j<i$, and setting $t'_{i,i+1} = \\eta(\\mu_{i+1})$;}\n \\item $\\mu'$ is $\\mu$ if $A'$ has the same bottom row as $A$; otherwise, $\\mu'$ is obtained from $\\mu$ by replacing {$\\mu_{i+1}$ with $\\eta(t_{i,i+1})$};\n \\item $\\q m' = (\\ldots,k_{i+1}-\\delta_{\\nwarrow}(\\mu'_{i+1})+1,k_{i+1}+\\delta_{\\nearrow}(\\mu'_{i+1}),\\ldots)$.\n\\end{itemize}\n{What remains is\n$$\n\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{i=1}^n (-1)^{n-i} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T),k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T)+1,\\ldots,x),\n$$\nand we can now apply $\\pi_{i,n} \\circ \\pi_{i,n} \\circ \\dots \\circ \\pi_{n-1,n}$ to obtain what is claimed.}\n\n\\emph{Construction of $\\Phi_4$.} By switching the order in which we do disjoint unions, we arrive at\n$$\\bigsqcup_{T \\in \\AP_{n}}  \\bigsqcup_{i=1}^n \\bigsqcup_{\\mu \\in \\AR_n} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots).$$\nLet us define a sijection $\\Lambda_{n,i} \\colon \\AR_n \\Rightarrow (\\{\\cdot\\},\\{\\})$. For $\\mu' = (\\nwarrow,\\ldots,\\nwarrow,\\nearrow,\\ldots,\\nearrow)$ (with $i$ $\\nwarrow$'s), take $\\Lambda_{n,i}(\\mu') = \\cdot$ and $\\Lambda_{n,i}(\\cdot) = \\mu'$. For every other $\\mu$, take the smallest $p$ so that $\\mu_p \\neq \\mu'_p$. If $p \\leq i$, replace $\\nearrow$ with $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ and vice versa in position $p$ to get $\\Lambda_{n,i}(\\mu)$ from $\\mu$, and if $p > i$, replace $\\nwarrow$ with $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ and vice versa in position $p$ to get $\\Lambda_{n,i}(\\mu)$ from $\\mu$. If $\\mu \\neq \\mu'$, $(\\delta_{\\nearrow}(\\mu_1),\\ldots,\\delta_{\\nearrow}(\\mu_{i-1}),\\delta_{\\nwarrow}(\\mu_{i+1}),\\ldots,\\delta_{\\nwarrow}(\\mu_n))$ are unaffected by this sijection, so it induces a sijection\n\\begin{multline*}\n  \\bigsqcup_{\\mu \\in \\AR_n} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots) \\\\\n  \\Rightarrow \\GT(\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i}(T),\\ldots).\n\\end{multline*}\nWe switch disjoint unions again, and we get\n$$\\bigsqcup_{i=1}^n \\bigsqcup_{T \\in \\AP_{n}}  \\GT(k_1+c_1(T),\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i}(T),\\ldots,k_{n} + c_{n-1}(T)).$$\nFor chosen $i$, use Proposition \\ref{prop:sijections} (3) for $\\psi = \\Psi_{n,i} \\circ \\Psi_{n,n}^{-1}$ and $\\varphi_t = \\id$. We get a sijection to\n$$\\bigsqcup_{i=1}^n \\bigsqcup_{T \\in \\AP_{n}}  \\GT(k_1+c_1(T),\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i+1}(T),\\ldots,k_{n} + c_{n}(T)).$$\nIf we switch disjoint unions one last time, we can use the sijection $\\tau^{-1}$ (see Problem \\ref{prob:tau}), and we get\n$$\\bigsqcup_{T \\in \\AP_n} \\GT(d(\\q k,T)) = \\SGT(\\q k).$$\nThis completes the construction of $\\Phi_4$ and therefore of $\\Phi$.\n\\end{proof}\n\n\\begin{problem}\n Given $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n $$\\Gamma = \\Gamma_{\\q k,x} \\colon \\MT(\\q k) \\Rightarrow \\SGT(\\q k).$$\n\\end{problem}\n\\begin{proof}[Construction]\n The proof is by induction on $n$. For $n = 1$, both sides consist of one (positive) element, and the sijection is obvious. Once we have constructed $\\Gamma$ for all lists of length less than $n$, we can construct $\\Gamma_{\\q k,x}$ as the composition of sijections\n $$\\MT(\\q k) \\stackrel{\\Xi_{\\q k}}{\\Longrightarrow} \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\MT(\\q l) \\stackrel{\\sqcup \\sqcup \\Gamma}{\\Longrightarrow} \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\stackrel{\\Phi_{\\q k,x}}{\\Longrightarrow} \\SGT(\\q k),$$\n where $\\sqcup \\sqcup \\Gamma$ means $\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\Gamma_{\\q l,x}.$\n\\end{proof}\n\nRunning the code shows that the main sijection $\\Gamma$ indeed depends on the choice $x$. As an example, take $\\q k = (1,2,3)$. In this case, $\\MT(\\q k)$ has $7$ positive elements, and $\\SGT(\\q k)$ has $10$ positive and $3$ negative elements. For $x = 0$, the sijection is given by\n$$\\mtthree 1 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 1 1 1 1,\\mttwo  \\searrow \\searrow \\searrow\\right) \\qquad \\mtthree 2 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 2 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right) \\qquad \\mtthree 1 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 2 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 3 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 2 3 2 2 3, \\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 2 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 3 1 2, \\mttwo  \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow\\right)$$\n$$\\mtthree 3 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 3 3 3 3 3,\\mttwo  \\swarrow \\swarrow \\swarrow\\right) \\qquad \\left(\\mtthree 2 2 2 2 2 3, \\mttwo \\swarrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\swarrow\\right)$$\n$$\\left(\\mtthree 2 2 2 2 3 1,\\mttwo \\searrow \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\right) \\qquad \\left(\\mtthree 2 2 2 1 2 2,\\mttwo \\searrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\swarrow\\right)$$\nwhile for $x = 1$, it is given by\n$$\\mtthree 1 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 1 1 1 1,\\mttwo  \\searrow \\searrow \\searrow\\right) \\qquad \\mtthree 2 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 3,\\mttwo \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 1 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 2 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 3 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 2 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 1 2 2,\\mttwo \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 3 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 3 3 3 3 3,\\mttwo  \\swarrow \\swarrow \\swarrow\\right) \\qquad \\left(\\mtthree 2 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\swarrow\\right)$$\n$$\\left(\\mtthree 2 2 2 3 1 2, \\mttwo  \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2, \\mttwo  \\swarrow \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\right) \\qquad \\left(\\mtthree 2 2 2 2 3 1,\\mttwo \\searrow \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2, \\mttwo  \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\swarrow\\right)$$\n\n\\section{Concluding remarks}\n\n\\subsection*{Future work}\n\nIn this article, we have presented the first bijective proof of the operator formula. The operator formula is the main tool for non-combinatorial proofs of several results where alternating sign matrix objects are related to plane partition objects, or simply for showing that $n \\times n$ ASMs are enumerated by \\eqref{asm}.\n\n\\begin{itemize}\n\\item The operator formula was used in \\cite{Fis07} to show that $n \\times n$ ASMs are counted by \\eqref{asm} and, more generally, to count ASMs with respect to the position of the unique $1$ in the top row.\n\\item While working on this project, we actually realized that the final calculation in \\cite{Fis07} also implies that ASMs are equinumerous with DPPs without having to use Andrews' result \\cite{And79} on the number of DPPs; more generally, we can even obtain the equivalence of the refined count of $n \\times n$ ASMs with respect to the position of the unique $1$ in the top row and the refined count of DPPs with parts no greater than $n$ with respect to the number of parts equal to $n$. This was conjectured in  \\cite{MilRobRum83} and first proved in \\cite{BehDifZin12}.\n\\item In \\cite{Fis19a}, the operator formula was used to show that ASTs with $n$ rows are equinumerous with TSSCPPs in a $2n \\times 2n \\times 2n$-box. Again we do not rely on Andrews' result \\cite{And94} on the number of TSSCPPs and we were actually able to deal with a refined count again (which has also the same distribution as the position of the unique $1$ in the top row of an ASM).\n\\item In \\cite{Fis19b}, we have considered alternating sign trapezoids (which generalize ASTs) and, using the operator formula, we have shown that they are equinumerous with objects generalizing DPPs. These objects were already known to Andrews and he actually enumerated them in \\cite{And79}. Later Krattenthaler \\cite{Kra06} realized that these more general objects are (almost trivially) equivalent to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. Again we do not rely on Andrews' enumeration of these generalized DPPs, and in this case we were able to include three statistics.\n\\end{itemize}\n\nWe plan to work on converting the proofs just mentioned into bijective proofs. For those mentioned in the first and second bullet point, this has already been worked out.\nThe attentive reader will have noticed that working out all of them will link all four known classes of objects that are enumerated by \\eqref{asm}.\n\n\\subsection*{Computer code}\n\nAs mentioned before, we consider computer code for the constructed sijections an essential part of this project. The code (in python) is available at \n\\begin{center}\n\\url{https:\/\/www.fmf.uni-lj.si\/~konvalinka\/asmcode.html}.\n\\end{center}\n All the constructed sijections are quite efficient.  If run with pypy, checking that $\\Gamma_{(1,2,3,4,5),0}$ is a sijection between $\\MT(1,2,3,4,5)$ (with $429$ positive elements and no negative elements) and $\\SGT(1,2,3,4,5)$ (with $18913$ positive elements and $18484$ negative elements) takes less than a minute. Of course, the sets involved can be huge, so checking that  $\\Gamma_{(1,2,3,4,5,6),0}$ is a sijection between $\\MT(1,2,3,4,5,6)$ (with $7436$ positive elements and no negative elements) and $\\SGT(1,2,3,4,5,6)$ (with $11167588$ positive elements and $11160152$ negative elements) took almost 20 hours.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nGiven a complex non-K\\\"ahler $n$-dimensional manifold $(\\rm{M},\\rm{J})$ it is a natural and meaningful problem to find special Hermitian metrics which might help in understanding the geometry of $\\rm{M}$. Great effort has been spent in the last decades in this research topic and among special metrics the pluriclosed and the balanced conditions have shown to be highly significant.\\par \nThe balanced condition can be defined saying that the fundamental form $\\o=h(\\cdot,\\rm{J}\\cdot)$ of a Hermitian metric $h$ satisfies the non-linear condition $d\\o^{n-1}=0$ or equivalently, $-\\rm{J} \\theta = \\d\\o=0$, where $\\d$ denotes the codifferential and $\\theta$ the torsion $1$-form (see e.g. \\cite{Ga2}).\nWhile this concept appears in \\cite{Ga1} under the name of semi-K\\\"ahler (see also \\cite{Gra}), in \\cite{Mi} the balanced condition was started to be thoroughly investigated, highlighting also the duality with the K\\\"ahler condition and establishing necessary and sufficient conditions for the existence of these metrics in terms of currents. While K\\\"ahler metrics are obviously balanced and share with these the important relation among Laplacians $\\Delta_{\\partial}=\\Delta_{\\overline\\partial}=\\frac 12 \\Delta$ (see \\cite{Ga1}), there are many examples of non-K\\\"ahler manifolds carrying balanced metrics. Basic examples are given by compact complex parallelizable manifolds, which are covered by complex unimodular Lie groups $\\rm{G}$ and every left invariant Hermitian metric turns out to be balanced (see \\cite{AG}\\cite{Ga1}\\cite{Gr}). Further examples of balanced metrics are provided by any Hermitian invariant metric on a compact homogeneous flag manifold (see also \\cite{FGV} for a characterization of compact homogeneous complex manifolds carrying balanced metrics) as well as by twistor spaces of certain self-dual $4$-manifolds (\\cite{Mi}) and more generally (\\cite{To}) by twistor spaces of compact hypercomplex manifolds (see also \\cite{Fo} for other examples on toric bundles over hyperk\\\"ahler manifolds). Contrary to the K\\\"ahlerness condition, being balanced is a birational invariant (see \\cite{AB1}, so that e.g. Moishezon manifolds are balanced) and compact complex manifolds $X$ which can be realized as the base of a holomorphic proper submersion $f:Y\\to X$ inherit the balanced condition whenever $Y$ has it (\\cite{Mi}), while the balanced property is not stable under small deformations of the complex structure (see \\cite{AB2},\\cite{FuY}, \\cite{AU}). On the other hand, the balanced condition is obstructed, as on compact manifolds with balanced metrics no compact complex hypersurface is homologically trivial, so that for instance Calabi-Eckmann manifolds do not carry balanced metrics. This is in contrast with the fact that Gauduchon metrics, which statisfy the weaker condition  $\\partial\\bar\\partial\\o^{n-1}=0$, always exist on a compact complex manifold.  \\par \nIn more recent years, the rising interest in the Strominger System (see \\cite{GF} and \\cite{FeY} for the case of invariant solutions on complex Lie groups) has given balanced metrics a really central role in non-K\\\"ahler geometry, as the equivalence between the dilatino equation (i.e. one of the equations of the system) and the conformally balanced equation requires the solutions of the system to be necessarily balanced. We refer also to the work \\cite{FLY}, where new examples of balanced metrics are constructed on some Calabi Yau non-K\\\"ahler threefolds, as well as to the results in \\cite{BV}, where a new balanced flow is introduced and investigated.  \\par \nThe main goal of this paper is to search for invariant special Hermitian, in particular balanced, metrics in the class of semisimple real non-compact Lie groups and on their compact (non-K\\\"ahler) quotients by a cocompact lattice; actually it appears that, despite invariant complex structures on semisimple (reductive) Lie algebras being fully classified in \\cite{Sn} (after the special case of compact Lie algebras had been considered by Samelson (\\cite{Sam}) and later in \\cite{Pi}), they have never been deeply investigated from this point of view. In contrast, the case of $\\K$ compact is fully understood, as in such a case it is very well known that every invariant complex structure can be deformed to an invariant one for which the opposite of the Cartan-Killing form is a pluriclosed Hermitian metric $h$, i.e. it satisfies $dd^c\\o_h=0$. Moreover it has been proved in \\cite{FGV} that $\\K$ does not carry {\\it any} balanced metric at all, fueling the conjecture (\\cite{FV}) that a compact complex manifold carrying two Hermitian metrics, one balanced and the other pluriclosed, must be actually K\\\"ahler.\\par\nMore specifically, in this work we focus on a large class of simple non-compact real Lie algebras $\\gg_o$ of even dimension, namely those which are of inner type, i.e. when the maximal compactly embedded subalgebra $\\mathfrak{k}$ in a Cartan decomposition of $\\gg_o$ contains a Cartan subalgebra. In these algebras we construct standard invariant complex structures (regular in \\cite{Sn}) and write down the balanced condition for invariant Hermitian metrics. A careful analysis of the resulting equation together with some general argument on root systems allows us to show the existence of a suitable invariant complex structure and a corresponding Hermitian metric satisfying the balanced equation. By Borel's Theorem, every semisimple Lie group $\\rm{G}_o$ admits a cocompact lattice $\\Gamma$ so that the compact quotient $\\rm{G}_o\/\\Gamma$ inherits the invariant balanced structure from $\\rm{G}_o$. We note here that the resulting metrics come in families and moreover the same kind of arguments can be applied to show the existence of balanced structures on quotients $\\rm{G}_o\/\\S$, where $\\rm{G}_o$ is any simple non-compact Lie group of inner type of any dimension and $\\S$ is a suitable abelian closed subgroup.\\par \nWe are also able to prove that the compact quotients $\\rm{M}=\\rm{G}_o\/\\Gamma$, endowed with the invariant complex structure that allows the existence of balanced metrics, do not carry any pluriclosed metric. This result is in accordance with the conjecture by Fino and Vezzoni and in some sense reflects a kind of duality between the compact and non-compact case, switching the existence of balanced\/pluriclosed Hermitian metrics. In the last section, we prove that these balanced manifolds $\\rm{M}$, despite having vanishing first Chern class, carry no non trivial holomorphic $(n,0)$-forms; furthermore  we prove that they have vanishing Chern scalar curvature. This last property may allow to better understand the geometry of these manifolds, according to some more recent results concerning the implications of vanishing Chern-scalar curvature on some geometric features (see \\cite{Y}). \\par\nThe paper is structured as follows. In Section 2, we review basic facts on simple real non-compact Lie algebras with invariant complex structures and we consider a class of invariant Hermitian metrics for which we write down the balanced condition in terms of roots. In section 3 we state our main result, namely Theorem \\ref{main}, and we prove it by means of several steps. We first rewrite the balanced equation in terms of simple roots and then the key Lemma \\ref{L1} allows us to select an invariant complex structure so that the relative balanced equation admits solutions. In section 4 we prove that the complex manifolds that we constructed in the previous section and that admit balanced metrics, do not carry {\\it any} pluriclosed metric. In the last section, we show in Theorem 5.1 that these complex compact manifolds $(\\rm{M},\\rm{J})$ have trivial first Chern class and that the balanced metrics we have constructed have vanishing Chern scalar curvature; as a consequence we show that the Kodaira dimension $\\kappa(\\rm{M})=-\\infty$.\n\n\n\\par \n\\vspace{0.5cm}\n{\\bf Aknowledgements.} The second author was supported by GNSAGA of INdAM and by the project PRIN 2017 ``Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics'', n. 2017JZ2SW5.\\par \nThe authors would like to thank Daniele Angella for valuable conversations.  \n\n\\section{Preliminaries}\n\nLet $\\gg_o$ be a real simple $2n$-dimensional Lie algebra. It is well known that either the complexification $\\gg_o^c$ is a complex simple Lie algebra (and in this case $\\gg_o$ is called absolutely simple) or $\\gg_o$ is the realification $\\gg_\\mathbb R$ of a complex simple Lie algebra $\\gg$ (see e.g. \\cite{He}). \\par \nWhen $\\gg_o$ is even dimensional, it is known (\\cite{Mo}, see also \\cite{Sas}) that $\\gg_o$ admits an invariant complex structure, namely an endomorphism $\\rm{J}\\in \\End(\\gg_o)$ with $\\rm{J}^2=-\\rm{Id}$ and vanishing Nijenhuis tensor or, equivalently, such that \n$$\\gg_o^c = \\gg_o^{10}\\oplus \\gg_o^{01},\\qquad [\\gg_o^{10},\\gg_o^{10}]\\subseteq \\gg_o^{10}.$$\nIf $\\rm{G}_o$ is any Lie group with Lie algebra $\\gg_o$, then the endomorphism $\\rm{J}$ defines a (left)-invariant complex structure on $\\rm{G}_o$. Moreover, thanks to a result due to Borel (\\cite{Bo}), there exists a discrete, torsionfree cocompact lattice $\\Gamma$ so that $\\rm{M}:= \\rm{G}_o\/\\Gamma$ is compact and the left-invariant complex structure $\\rm{J}$ on \n$\\rm{G}_o$ descends to a complex structure $\\rm{J}$ on $\\rm{M}$.\\par \nWe recall that when $\\rm{G}_o$ is compact and even-dimensional, i.e. $\\gg_o$ is of compact type, the existence of an invariant complex structure was already established by Samelson (\\cite{Sam}), while in \\cite{Pi} it was shown that every invariant complex structure on $\\rm{G}_o$ is obtained by means of Samelson's construction. \\par\nIf we now consider an even-dimensional $\\rm{G}_o$ and a compact quotient $\\rm{M}$ endowed with an invariant complex structure $\\rm{J}$, we are interested in the existence of special Hermitian metrics $h$. The following proposition states a known fact, namely the non-existence of (invariant) K\\\"ahler structures.\n\\begin{prop}\\label{invK} The group $\\rm{G}_o$ does not admit any invariant K\\\"ahler metric and the compact quotient $\\rm{M}=\\rm{G}_o\/\\Gamma$ is not K\\\"ahler.\n\\end{prop} \n\\begin{proof} The first assertion is contained in \\cite{Ch}, but we give here an elementary proof. If $\\o$ is an invariant symplectic form \non $\\gg_o$, then the closedness condition $d\\o=0$ can be written as follows for $x,y,z\\in\\gg_o$  \n\\begin{equation}\\label{closed}\\o([x,y],z)+\\o([z,x],y) + \\o([y,z],x)=0.\\end{equation}\nIf $B$ denotes the non-degenerate Cartan-Killing form of $\\gg_o$, then we can define the endomorphism $F\\in\\End(\\gg_o)$ by $B(Fx,y)=\\o(x,y)$ ($x,y\\in\\gg_o$) so that $F$ turns out to be a derivation by \\eqref{closed}. As $\\gg_o$ is semisimple, there exists a unique $z\\in \\gg_o$ with $F=\\ad(z)$, so that $z\\in \\ker\\o$, a  contradiction.  \\par\nWe now suppose that the compact manifold $\\rm{M}$ has a K\\\"ahler metric with K\\\"ahler form $\\o$. Using $\\o$ and a symmetrization procedure that goes back to \\cite{Be}, we now construct an {\\it invariant} K\\\"ahler form on $\\rm{G}_o$, obtaining a contradiction. We fix a basis $x_1,...,x_{2n}$ of $\\gg_o$ and we extend each vector as a left invariant vector fields on $G_o$; these vector fields can be  projected down to $M$ as vector fields $x_1^*,\\ldots,x_{2n}^*$ that span the tangent space $TM$ at each point. As $\\rm{G}_o$ is semisimple, we can find a biinvariant volume form $d\\mu$, that also descends to a volume form on $\\rm{M}$. We now define a left-invariant non-degenerate $2$-form $\\phi$ on $G_o$ by setting\n$$\\phi_e(x_{i},x_j) := \n\\int_M \\o(x_{i}^*,x_{j}^*)\\ d\\mu.$$\nAs $\\mathcal L_{x_k^*}d\\mu = 0$ for every $k$, we have for every $i,j,k=1,\\ldots,2n$\n$$\\int_M x_k^*\\o(x_{i}^*,x_{j}^*)\\ d\\mu = \\int_M \\mathcal L_{x_k^*}( \\o(x_{i}^*,x_{j}^*)\\ d\\mu) = 0$$ by Stokes' theorem and therefore we obtain that \n$$d\\phi(x_{i},x_j,x_k) = \n\\int_M d\\o(x_{i}^*,x_j^*,x_{k}^*)\\ d\\mu = 0.$$\nThis implies that $\\phi$ is a symplectic form and the proof is concluded.\\end{proof}\nTherefore we are interested in the existence of special Hermitian metrics on the complex manifold ($\\rm{M},\\rm{J}$), in particular balanced and pluriclosed metrics, when the group $\\rm{G}_o$ is of non-compact type. \\par \nThe case of a simple Lie algebra $\\gg_o$ which is the realification of a complex simple Lie algebra $\\gg$ can be easily treated and will be dealt with in subsection 2.3.\\par\nWe will now focus on some subclasses of simple real algebras, namely those which are absolutely simple and of inner type. \\par\n\\subsection{Simple Lie algebras of inner type}\\label{simple} Let $\\gg_o$ be an absolutely simple real algebra of non-compact type. It is well-known that $\\gg_o$ admits a Cartan decompositon \n$$\\gg_o = \\mathfrak{k} + \\mathfrak{p},$$\nwhere $\\mathfrak{k}$ is a maximal compactly embedded subalgebra and \n$$[\\mathfrak{k},\\mathfrak{p}]\\subseteq \\mathfrak{p},\\quad [\\mathfrak{p},\\mathfrak{p}]\\subseteq \\mathfrak{k},$$\nso that $(\\gg_o,\\mathfrak{k})$ is a symmetric pair. Moreover the algebra $\\gg_o$ is said to be {\\it of inner type} when the symmetric pair $(\\gg_o,\\mathfrak{k})$ is of inner type, i.e. when a Cartan subalgebra $\\mathfrak{t}$ of $\\mathfrak{k}$ is a Cartan subalgebra of $\\gg_o$, i.e. its complexification $\\mathfrak{t}^c$ is a Cartan subalgebra of $\\gg_o^c$. Using the notation as in \\cite{He}, p.~126, we obtain the list of all inner symmetric pairs $(\\gg_o,\\mathfrak{k})$ of non-compact type with $\\gg_o$ simple and even dimensional (Table 1).  \\par\n\\begin{table}[ht]\\label{T1}\n\t\\centering\n\t\\renewcommand\\arraystretch{1.1}\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t{\\mbox{Type}}\t\t\t\t& \t$\\gg$\t\t\t\t\t&\t$\\mathfrak{k}$\t& \t{\\mbox{conditions}}\t\t\t \t\\\\ \\hline \\hline\n\t$A$ &\t$\\mathfrak{su}(p,q)$ & $\\mathfrak{su}(p) + \\mathfrak{su}(q) +\\mathbb R$ & $p\\geq q\\geq 1$, \\ $p+q\\ {\\rm{odd}}$\t\t\t\t\\\\ \\hline\n\t$B$ & $\\mathfrak{so}(2p+1,2q)$ & $\\mathfrak{so}(2p+1) + \\mathfrak{so}(2q)$ & $p\\geq 0,q\\geq 1$, \\ $p+q\\ {\\rm{even}}$\t\t\t\\\\ \\hline\n\t$C$ & $\\mathfrak{sp}(2n,\\mathbb R)$ & $\\mathfrak{su}(2n)+\\mathbb R$ & $n\\geq 1$\t\t\t\\\\ \\hline\n\t$C$ & $\\mathfrak{sp}(p,q)$ & $\\mathfrak{sp}(p) + \\mathfrak{sp}(q)$ & $p,q\\geq 1$, \\ $p+q\\ {\\rm{even}}$\t\t\t\\\\ \\hline\n\t$D$ & $\\mathfrak{so}(4n)^*$ & $\\mathfrak{su}(2n)+\\mathbb R$ & $n\\geq 2$\t\t\t\\\\ \\hline\n\t$D$ & $\\mathfrak{so}(2p,2q)$ & $\\mathfrak{so}(2p) + \\mathfrak{so}(2q)$ & $p,q\\geq 1,p+q\\ \\rm{even}\\ \\geq 4$\t\t\t\\\\ \\hline\n\t$G$ & $\\gg_{2(2)}$ & $\\mathfrak{su}(2)+\\mathfrak{su}(2)$ & \t\t\t\\\\ \\hline\n\t$F$ & $\\mathfrak{f}_{4(-20)}$ & $\\mathfrak{so}(9)$ & \t\t\t\\\\ \\hline\n\t$F$ & $\\mathfrak{f}_{4(4)}$ & $\\mathfrak{su}(2)+\\mathfrak{sp}(3)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{6(2)}$ & $\\mathfrak{su}(2)+\\mathfrak{su}(6)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{6(-14)}$ & $\\mathfrak{so}(10)+\\mathbb R$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{8(8)}$ & $\\mathfrak{so}(16)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{8(-24)}$ & $\\mathfrak{su}(2)+\\mathfrak{e}_7$ & \t\t\t\\\\ \\hline\n\t\\end{tabular}\n\t\\vspace{0.1cm}\n\t\\caption{Inner symmetric pairs $(\\gg,\\mathfrak{k})$ of non-compact type with $\\gg$ simple and even dimensional.}\\label{table}\n\\end{table}\n\n\\subsection{Invariant complex structures} In this section we will describe how to construct invariant complex structures on even-dimensional absolutely simple non-compact Lie algebras $\\gg_o$. \\par\nWe fix a maximal abelian subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$, so that $\\mathfrak{h}:= \\mathfrak{t}^c$ is a Cartan subalgebra of $\\gg:= \\gg_o^c$. Note that if $\\gg_o$ is even dimensional , the same holds for $\\mathfrak{t}$. The corresponding root system is denoted by $R$ and we have the following decompositions \n$$\\mathfrak{k}^c = \\mathfrak{t}^c \\oplus \\bigoplus_{\\a\\in R_\\mathfrak{k}}\\gg_\\a,\\quad \\mathfrak{p}^c = \\bigoplus_{\\a\\in R_\\mathfrak{p}}\\gg_\\a,$$\nwhere a root $\\a$ will be called {\\it compact} (resp. {\\it non-compact}), when $\\gg_\\a\\subseteq \\mathfrak{k}^c$ (resp. $\\gg_\\a\\subseteq \\mathfrak{p}^c$) and the set of all compact (resp. non-compact) roots is denoted by $R_\\mathfrak{k}$ (resp. $R_\\mathfrak{p}$). It is a standard fact that $\\mathfrak{u}:= \\mathfrak{k} + i\\mathfrak{p}\\subseteq \\gg$ is a compact real form of $\\gg$ and that we can choose a basis $\\{E_\\a\\}_{\\a\\in R}$ of root spaces so that \n$$\\tau(E_\\a) = -E_{-\\a}, \\qquad B(E_\\a,E_{-\\a}) = 1,\\qquad [E_\\a,E_{-\\a}] = H_\\a$$\nwhere $\\tau$ denotes the anticomplex involution defining $\\mathfrak{u}$, $B$ is the Cartan Killing form of $\\gg$ and $H_\\a$ is the $B$-dual of $\\a$ (see e.g. \\cite{He}). If $\\sigma$ is the involutive anticomplex map defining $\\gg_o$, we then have that \n$$\\sigma(E_\\a) = -E_{-\\a},\\quad \\a\\in R_\\mathfrak{k},$$\n$$\\sigma(E_\\a) = E_{-\\a},\\quad \\a\\in R_\\mathfrak{p}.$$\nIf we fix an ordering , namely a splitting $R = R^+\\cup R^-$ with $R^-=-R^+$ and $(R^++R^+)\\cap R \\subseteq R^+$, we can define a subalgebra \n$$\\mathfrak{q} := \\mathfrak{h}_1 \\oplus \\bigoplus_{\\a\\in R^+}\\gg_\\a,$$\nwhere $\\mathfrak{h}_1\\subset \\mathfrak{h}$ is a subspace so that $\\mathfrak{h}_1\\oplus \\sigma(\\mathfrak{h}_1)= \\mathfrak{h}$. The so defined subalgebra $\\mathfrak{q}\\subset \\gg$ satisfies \n$$\\gg = \\mathfrak{q} \\oplus \\sigma(\\mathfrak{q})$$\nand therefore it defines a complex structure $\\rm{J}$ on $\\gg_o$ with the property that $\\mathfrak{q} = \\gg_o^{10}$. This complex structure depends on the arbitrary choice of $\\mathfrak{h}_1$, i.e. on the arbitrary choice of a complex structure on $\\mathfrak{t}$. \\par \nWe remark that the complex structure $\\rm{J}$ enjoys the further property of being $\\ad(\\mathfrak{t})$-invariant, namely \n$$[\\ad(x),\\rm{J}] = 0,\\quad x\\in\\mathfrak{t}.$$\nTherefore if $\\rm{G}_o$ is a Lie group with Lie algebra $\\gg_o$, then $\\rm{J}$ extends to a left-invariant complex structure on $\\rm{G}_o$ and it will be also right-invariant with respect to right translations by elements $h\\in {\\rm{T}}:= \\exp(\\mathfrak{t})$ (note that ${\\rm{T}}$ might be non-compact, unless $\\rm{G}_o$ has finite center). \\par\nWe will call such an invariant complex structure {\\it standard}.\n\\begin{remark} In \\cite{Sn} the class of (simple) real Lie algebras of inner type is called ``Class I'' and it is then proved that {\\it every} invariant complex structure in these algebras are standard, with respect to a suitable choice of a Cartan subalgebra (such complex structures are called regular in \\cite{Sn}).\n\\end{remark}\n\n\\subsection{Invariant metrics and the balanced condition}\\label{inv}\nLet $\\rm{M}$ be a compact complex manifold of the form $\\rm{G}_o\/\\Gamma$, endowed with a complex structure $\\rm{J}$ which is induced by a standard invariant complex structure $\\rm{J}$ on $\\rm{G}_o$, as in the previous section. It is clear that any left invariant $\\rm{J}$-Hermitian metric $h$ on $\\rm{G}_o$ induces an Hermitian metric $\\bar h$ on $\\rm{M}$ and $\\bar h$ is balanced or pluriclosed if and only if $h$ is so. For the converse, we prove the following\n\\begin{prop} If $(\\rm{M},\\rm{J})$ admits  a balanced (pluriclosed) Hermitian metric, there exists a left invariant and right $\\T$-invariant Hermitian metric on $\\rm{G}_o$ which is balanced (pluriclosed resp.).  \\end{prop}\n\\begin{proof} Suppose we have a balanced metric $h$ on $M$ with associated fundamental form $\\o$. Then using the same notation and arguments as in the proof of Prop.\\ref{invK}, we define a left-invariant positive $(n-1,n-1)$-form $\\phi$ on $G_o$ as follows\n$$\\phi_e(x_{i_1},\\ldots,x_{i_{2n-2}}) := \n\\int_M \\o^{n-1}(x_{i_1}^*,\\ldots,x_{i_{2n-2}}^*)\\ d\\mu.$$\nAs $d\\o^{n-1}=0$, we obtain that also $d\\phi=0$. Therefore, we\ncan find an unique $(1,1)$-form $\\hat \\o$ so that $\\hat\\o^{n-1} = \\phi$ (see \\cite{Mi}) and the metric given by $\\hat \\o$ is balanced. As $\\phi$ is left invariant, so is $\\hat \\omega$ by uniqueness. Now, the group $\\Ad(\\T)$ is compact and using a standard avaraging process we can make $\\phi_e$ also $\\Ad(\\T)$-invariant. This means that $\\phi$ is also invariant under right $\\T$-translations. Again, by the uniqueness, the same will hold true for $\\hat\\omega$.\\par As for the pluriclosed condition, the lifted metric from $\\rm{M}$ to $\\rm{G}_o$ is clearly pluriclosed and can be made $T$-invariant by a standard averaging. \\end{proof}\n\\begin{remark} We can now deal with the case when $\\gg_o$ is the realification of a simple Lie algebra $\\gg$. In this case the complex structure $\\rm{J}$ commutes with $\\ad(\\gg_o)$ and $\\gg_o = \\mathfrak{u} + i\\mathfrak{u}$ is a Cartan decomposition, where $\\mathfrak{u}$ is a compact real form of $\\gg$. Let $\\rm{G}_o$ be a real group with algebra $\\gg_o$ and let $\\U$ be the compact subgroup with algebra $\\mathfrak{u}$. Then the metric $h$ which coincides with $-B$ on $\\mathfrak{u}$, with $B$ on $i\\mathfrak{u}$ and such that $h(\\mathfrak{u},i\\mathfrak{u})=0$ is a Hermitian metric which is balanced. Indeed, $h$ is $\\Ad(\\U)$-invariant and therefore the corresponding $\\d\\o$ is  $\\Ad(\\U)$-invariant $1$-form, hence it vanishes identically. This is consistent with the fact that complex parallelizable manifolds carry balanced metrics as they carry Chern-flat metrics, as noted in  \\cite{Ga1}, p. 121 (see also \\cite{AG},\\cite{Gr}). \\par \nOn the other hand, $\\rm{G}_o$ admits no invariant pluriclosed metric. Indeed, any such metric $h$ can be avaraged to produce an $\\Ad(\\U)$-invariant pluriclosed metric, which would be balanced by the previous argument. This is not possible, as a metric which is balanced and pluriclosed at the same time has to be K\\\"ahler (see e.g. \\cite{AI}), contrary to Prop \\ref{invK}.\\end{remark}\n\nWe now focus on the case where $\\gg_o$ is absolutely simple of inner type, endowed with an invariant complex structure.  \nWe fix a Cartan subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$ with corresponding root system $R=R_\\mathfrak{k} \\cup R_\\mathfrak{p}$ as in section \\ref{simple} and we consider an ordering \n$R = R^+\\cup R^-$ giving an invariant complex structure $\\rm{J}_o$ on $\\gg_o\/\\mathfrak{t}$. We extend $\\rm{J}_o$ to an invariant complex structure $\\rm{J}$ on $\\gg_o$. \\par \nWe also fix a basis of a complement of $\\mathfrak{t}$ in $\\gg_o$\n$$v_\\a := \\frac 1{\\sqrt 2}(E_\\a-E_{-\\a}),\\ \nw_\\a := \\frac i{\\sqrt 2}(E_\\a+E_{-\\a}),\\ \\a\\in R_\\mathfrak{k}^+,$$\n$$v_\\a := \\frac 1{\\sqrt 2}(E_\\a+E_{-\\a}),\\ \nw_\\a := \\frac i{\\sqrt 2}(E_\\a-E_{-\\a}),\\ \\a\\in R_\\mathfrak{p}^+,$$\nso that $v_\\a,w_\\a\\in \\gg_o$ for every $\\a\\in R^+$ and moreover \n$$Jv_\\a = w_\\a, \\ Jw_\\a = -v_\\a,$$\n$$[H,v_\\a] = -i\\a(H)w_\\a,\\ H\\in\\mathfrak{h},$$ \n$$[v_\\a,w_\\a] = iH_\\a,\\ \\a\\in R_\\mathfrak{k}^+,$$\n$$[v_\\a,w_\\a] = -iH_\\a,\\ \\a\\in R_\\mathfrak{p}^+.$$\n\n\nWe now construct invariant Hermitian metrics $h$ on $\\gg_o$. First, we define $h$ on $\\mathfrak{t}$ by choosing a $J$-Hermitian metric $h_\\mathfrak{t}$ on $\\mathfrak{t}$. If we set $\\mathfrak{m}_\\a := {\\rm{Span}}\\{v_\\a,w_\\a\\}_{\\a\\in R^+}$, we define for $\\a\\neq \\b\\in R^+$\n$$h(\\mathfrak{t},\\mathfrak{m}_\\a)= 0,\\quad h(\\mathfrak{m}_\\a,\\mathfrak{m}_\\b)= 0,$$\n$$h(v_\\a,v_\\a) = h(w_\\a,w_\\a) = h_\\a^2,\\quad h(v_\\a,w_\\a) = 0$$\nfor $h_a\\in \\mathbb R^+$.\\par\nIn particular we are interested in constructing balanced Hermitian metrics, namely Hermitian metrics whose associated $(1,1)$-form $\\o=h(\\cdot,\\rm{J}\\cdot)$ satisfies $d\\o^{n-1}=0$ or equivalently $\\d\\o=0$, where $\\d$ denotes the codifferential. \\par \nWe use the expression \n$$\\d\\o (x) = -{\\rm{Tr}}\\nabla_{\\cdot}\\o(\\cdot,x) = - \\sum_{i}^{2n}\\nabla_{e_i}\\o(e_i,x) = $$\n$$= \\sum_i \\o(\\nabla_{e_i}e_i,x) + \\o(e_i,\\nabla_{e_{i}}x),$$\nwhere $\\nabla$ denotes the Levi Civita connection of $h$ and $\\{e_i\\}$ is an orthonormal basis of $\\gg_o$ w.r.t. $h$. Note that both $h$ and $\\rm{J}$ are $\\ad(\\mathfrak{t})$-invariant and therefore $\\d\\o$, which is also $\\ad(\\mathfrak{t})$-invariant, does not vanish only when evaluated on elements $x\\in \\mathfrak{t}$.\\par We have the following expression for the Levi Civita connection, namely for $x,y,z\\in \\gg_o$ \n$$2h(\\nabla_xy,z) = h([x,y],z) + h([z,x],y) + h([z,y],x).$$\nThen for every $x\\in\\mathfrak{t}$, $y\\in \\gg_o$ \n$$h(\\nabla_{y}y,x) = h([x,y],y) = 0.$$\nTherefore for $x\\in\\mathfrak{t}$ we have\n\\begin{equation}\\label{comp}\\d\\o(x) = \\sum_i\\o(e_i,\\nabla_{e_i}x) = -\\sum_ih(Je_i,\\nabla_{e_i}x) = \\end{equation}\n$$=-\\frac 12\\left( h([e_i,x],Je_i)+ \n           h([Je_i,e_i],x) + h([Je_i,x],e_i)\\right).$$\nWe now observe that $J$ is $\\ad(\\mathfrak{t})$-invariant and therefore \n$h([Je_i,x],e_i) = -h([e_i,x],Je_i)$ for every $i=1,\\ldots,2n$, \nso that \\eqref{comp} can be written as \n$$-\\d\\o(x) = \\frac 12 \\sum_ih([Je_i,e_i],x)= $$\n$$= \\frac 12 \\cdot 2 \\left(\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2} h([w_\\a,v_\\a],x)+ \n\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}h([w_\\a,v_\\a],x)\\right)=$$\n$$= \\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}h(-iH_\\a,x) + \n\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}h(iH_\\a,x),$$\nso that \n$\\d\\o|_{\\mathfrak{t}} =0$ if and only if \n$$-\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}H_\\a + \\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}H_\\a=0.$$\nSumming up, the metric $h$ is balanced when the following equation is satisfied\n\\begin{equation}\\label{eq}\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}\\a = \\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}\\a.\\end{equation}\\par\\medskip\nNote that this does {\\it not} depend on the choice of the metric along the toral part $\\mathfrak{t}$.\n\n\\section{Main result}\\label{proof}\nIn this section we will prove our main result\n\\begin{theorem}\\label{main} Every non-compact simple Lie group $\\rm{G}_o$ of even dimension and of inner type admits an invariant complex structure $\\rm{J}$ and an invariant balanced $\\rm{J}$-Hermitian metric.\\end{theorem}\nNote that by Borel's Theorem, we can use a cocompact latice $\\Gamma\\subset\\rm{G}_o$ to obtain compact quotients $\\rm{M}=\\rm{G}_o\/\\Gamma$, which will inherit the same balanced structure. \\par \n\n\nWe start noting that equation \\eqref{eq} involves the unknowns $\\{h_\\a\\}_{\\a\\in R^+}$ and also a choice of positive roots, i.e. an ordering or equivalenty a complex structure on $\\gg_o$. We will always fix a complex structure on $\\mathfrak{t}$ once for all. It is known that giving an ordering on the root system $R$ is equivalent to the choice of a system of simple roots $\\Pi$ and that two systems of simple roots are conjugate under the action of the Weyl group $W$. We may fix a system of simple roots $\\Pi = \\{\\a_1,\\ldots,\\a_r\\}$ and put \n$\\Pi = \\Pi_c \\cup \\Pi_{nc}$, where $\\Pi_{c\/nc}$ denotes the set of simple roots which are compact or noncompact. We set $\\Pi_c=\\{\\phi_1,\\ldots,\\phi_k\\}$, $\\Pi_{nc}=\\{\\psi_1,\\ldots,\\psi_l\\}$, $k+l=r = {\\rm{rank}}(\\gg_o)$.\nEach root $\\a\\in R^+$ can be written as \n$$\\a = \\sum_{i=1}^k n_i(\\a)\\phi_i + \\sum_{j=1}^l m_j(\\a)\\psi_j$$\nfor $n_i(\\a),m_j(\\a)\\in\\mathbb N$ nonnegative integers. If we set $g_\\a:= \\frac 1{h_\\a^2}$ and $g_j:= g_{\\phi_j}, h_j := g_{\\psi_j}$, equation \\eqref{eq} can be written as \n$$\\sum_{\\a\\in R_\\mathfrak{k}^+,\\a\\not\\in\\Pi} g_\\a\\left(\\sum n_j(\\a)\\phi_j + \\sum_j m_j(\\a)\\psi_j\\right) + \\sum_j g_j \\phi_j = $$\n$$= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\a\\not\\in\\Pi} g_\\a\\left(\\sum n_j(\\a)\\phi_j + \\sum_j m_j(\\a)\\psi_j\\right) + \\sum_j h_j \\psi_j,$$\nand therefore \n\\begin{equation}\\label{sys1}\n\\left\\{\\begin{aligned}\ng_j &= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(\\a) &- \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(a),{}\\ \\  &j=1,\\ldots,k,\\\\\nh_j &= \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(\\a) &- \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),{}\\ \\  &j=1,\\ldots,l.\n\\end{aligned}\\right.\\end{equation}\n\\begin{remark} If we consider for instance the case $\\gg_o=\\mathfrak{su}(p,q)$ ($p+q$ even, $p,q\\geq 2$) and the standard system of simple roots $\\Pi=\\{\\epsilon_1-\\epsilon_2,\\epsilon_2-\\epsilon_3,\\ldots,\\epsilon_{p-1}-\\epsilon_p,\\epsilon_p-\\epsilon_{p+1},\\ldots,\\epsilon_{p+q-1}-\\epsilon_{p+q}\\}$ of $\\sl(p+q,\\mathbb C)$, then $\\Pi_{nc}=\\{\\epsilon_p-\\epsilon_{p+1}\\}$ and $\\Pi_c$ gives a system of simple roots for the semisimple part $\\mathfrak{k}_{ss}$ of $\\mathfrak{k}$. This means that every root $\\a\\in R_\\mathfrak{k}^+,\\a\\not\\in \\Pi$\nis a linear combination of roots in $\\Pi_c$ and therefore the righthandside of the last equation in \\eqref{sys1} is non-positive, so that \\eqref{sys1} has no solution. This shows that the choice of the invariant complex structure might not be straightforward.   \n\\end{remark}\n\nThe following lemmata provide key tools in our argument.\n\\begin{lemma}\\label{L1} For each symmetric pair $(\\gg_o,\\mathfrak{k})$ as in Table 1, $(\\gg_o,\\mathfrak{k})\\not\\cong (\\mathfrak{so}(1,2n),\\mathfrak{so}(2n))$ and given a Cartan subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$ with corresponding root system $R$, there exists an ordering of the roots, hence a system of simple roots $\\Pi$, such that \n\\begin{equation}\\label{prop} \\forall \\psi \\in \\Pi_{nc}\\ \\exists \\psi'\\in \\Pi_{nc}\\ {\\rm{with}}\\ \\psi+\\psi'\\in R.\\end{equation} \n\\end{lemma}\nThis implies that, if $\\Pi_{nc}=\\{\\psi_1,\\ldots,\\psi_l\\}$, then for every $\\psi_j\\in \\Pi_{nc}$ there exists $\\a\\in R_\\mathfrak{k}^+$ with $m_j(\\a)\\neq 0$ and $\\a\\in {\\rm{Span}}\\{\\Pi_{nc}\\}$.\\par\\medskip\n\\noindent {\\bf Remark} Note that $\\sp(1,1)\\cong\\mathfrak{so}(1,4)$ is also not admissible in the above Lemma. In general, for $\\gg_o = \\mathfrak{so}(1,2n)$ we have the standard system $\\Pi=\\{\\epsilon_i-\\epsilon_{i+1},\\epsilon_n,\\ i=1,\\ldots,n-1\\}$ with \n$\\Pi_{nc}= \\{\\epsilon_n\\}$. As $R_c$ consists precisely of all the short roots, it is clear that for any element $\\sigma$ of the Weyl group $W\\cong \\mathbb Z_2^n\\ltimes \\mathcal S_n$ we have that $\\sigma(\\Pi)_{nc}$ consists of one element. We will deal with this case later on.\n\\begin{proof} We first deal with the classical case. We start with the standard system of simple roots $\\Pi$, following the notation as in \\cite{He}. It is immediate to check that in this case $\\Pi_{nc}$ consists of a single root $\\psi$. \\par \nWe first deal with the case where $\\psi$ is a short root. Let $\\Lambda$ be the set of all simple roots which are connected to $\\psi$ in the Dynkin diagram relative to $\\Pi$. If $s\\in W$ denotes the reflection around $\\psi$, then $s$ leaves every element $\\Pi\\setminus \\Lambda$ pointwise fixed. We observe that $\\Lambda$ consists of either at most three short roots or it contains a long root.  In the first case, $s(\\Lambda) = \\{\\psi + \\lambda|\\ \\l\\in\\Lambda\\}\\subseteq R_\\mathfrak{p}$ so that $s(\\Pi)_{nc} = \\{-\\psi,s(\\Lambda)\\}$ and therefore the system of simple roots $s(\\Pi)$ satisfies \\eqref{prop}. If $\\Lambda$ contains a long root, then it also contains a short root, unless $(\\gg_o,\\mathfrak{k})=(\\mathfrak{so}(2,3),\\mathbb R + \\mathfrak{so}(3))$, that is isomorphic to $(\\sp(2),\\mathfrak{u}(2))$; this case will be dealt with in the second part of the proof.  Therefore $\\Lambda = \\{\\phi_1,\\phi_2\\}$ with $\\phi_1$ short and $\\phi_2$ long. Again the reflection $s$ around $\\psi$ gives $s(\\phi_1) = \\psi+\\phi_1$ and $s(\\phi_2) = \\phi_2+2\\psi\\in R_\\mathfrak{k}$ or $s(\\phi_2) = \\psi+\\phi_2\\in R_\\mathfrak{p}$. This implies that the system of simple roots $s(\\Pi)$ has $s(\\Pi)_{nc} = \\{-\\psi,\\psi+\\phi_1\\}$ or $\\{-\\psi,\\psi+\\phi_1,\\psi+\\phi_2\\}$ and in both cases it satisfies \\eqref{prop}.\\par \nWe are left with the case where $\\psi$ is a long root, namely the case where $\\gg_o=\\sp(2n,\\mathbb R)$ and $\\mathfrak{k} = \\mathfrak{u}(2n)$. A standard system of simple roots is given by $\\Pi=\\{\\epsilon_1-\\epsilon_2,\\epsilon_2-\\epsilon_3,\\ldots,\\epsilon_{2n-1}-\\epsilon_{2n},2\\epsilon_{2n}\\}$ and $\\Pi_{nc}= \\{\\psi=2\\epsilon_{2n}\\}$. Again using $s_{\\b}$, we see that $s_\\b(\\Pi)_{nc}=\\{-2\\epsilon_{2n},\\epsilon_{2n-1}+\\epsilon_{2n}\\}$ so that condition \\eqref{prop} is satisfied.\\par\nWe may now deal with the exceptional cases. Starting with the standard system of simple roots $\\Pi$, we list the set $\\Pi_{nc}$, that turns out to consist of a single root $\\b$.  For each case, using the symmetry $s_\\b$ we obtain  the system of simple roots $\\Pi':=s_\\b(\\Pi)$ that satisfies condition \\eqref{prop}.\\par \n\\noindent (1)\\ $(\\gg_o,\\mathfrak{k}) = (\\gg_2,\\mathfrak{su}(2)+\\mathfrak{su}(2))$. Here $\\Pi=\\{\\a,\\b\\}$, with $\\b$ long. We have $\\Pi_{nc}=\\{\\b\\}$ and $\\Pi'=\\{-\\b,\\a+\\b\\}$.\\par\n\\noindent (2)\\ $(\\gg_o,\\mathfrak{k}) = (\\mathfrak{f}_{4(-20)},\\mathfrak{so}(9))$. According to \\cite{He}, the standard system of simple roots is $\\Pi=\\{\\a_1=\\epsilon_2-\\epsilon_3,\\a_2=\\epsilon_3-\\epsilon_4,\\a_3=\\epsilon_4,\\a_4=\\frac 12(e_1-\\epsilon_2-\\epsilon_3-\\epsilon_4)\\}$ so that \n$\\Pi_{nc}=\\{\\a_4\\}$ and therefore  $\\Pi'_{nc}=\\{-\\a_4,\\a_4+\\a_3\\}$.\\par \n\\noindent (3)\\ $(\\gg_o,\\mathfrak{k}) = (\\mathfrak{f}_{4(4)},\\mathfrak{su}(2)+\\mathfrak{sp}(3))$. In this case $\\Pi_{nc}=\\{\\a_1\\}$ and therefore $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_2\\}$. \\par\n\\noindent\\ (4)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{8(8)},\\mathfrak{so}(16))$. For $\\mathfrak{e}_8$ we have the standard system of simple roots \n$$\\a_1=\\frac 12(\\epsilon_1+\\epsilon_8)-\\frac 12(\\epsilon_2+\\epsilon_3+\\epsilon_4+\\epsilon_5+\\epsilon_6+\\epsilon_7), \\a_2=\\epsilon_1+\\epsilon_2, $$ \n$$\\a_j=\\epsilon_{j-1}-\\epsilon_{j-2},\\ j=3,\\ldots,8.$$\nThen $\\Pi_{nc}=\\{\\a_1\\}$ and $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_3\\}$.\\par \n\\noindent\\ (5)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{8(-24)},\\mathfrak{su}(2)+\\mathfrak{e}_7)$. Keeping the same notation for simple roots as above, we have $\\Pi_{nc} = \\{\\a_8\\}$ and $\\Pi'_{nc}=\\{-\\a_8,\\a_8+\\a_7\\}$.\\par\n\\noindent\\ (6)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{6(2)},\\mathfrak{su}(2)+\\mathfrak{su}(6))$. As the system root system $\\Pi$ can be taken to be composed of the simple roots $\\{\\a_1,\\ldots,\\a_6\\}$ of $\\mathfrak{e}_8$, we have \n$\\Pi_{nc}=\\{\\a_2\\}$ and $\\Pi'_{nc}=\\{-\\a_2,\\a_2+\\a_4\\}$.\\par \n\\noindent\\ (7)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{6(-14)},\\mathbb R+\\mathfrak{so}(10))$. We have $\\Pi_{nc}=\\{\\a_1\\}$ and $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_3\\}$.\\end{proof}\n\n\n\\begin{lemma}\\label{L2} For every system of simple roots $\\Pi = \\Pi_{c}\\cup \\Pi_{nc}$ \n with $\\Pi_{c}=\\{\\phi_1,\\ldots,\\phi_k\\}$ we have \n$$\\forall\\ j=1,\\ldots,k,\\ \\exists\\ \\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in \\Pi : \\ n_j(\\a)\\neq 0,$$\n where $n_j(\\a)$ denotes the coordinate of $\\a$ along the root $\\phi_j$.\n\\end{lemma}\n\\begin{proof} We start noting that the centralizer $C_{\\mathfrak{k}^c}(\\mathfrak{p}^c) = C_{\\mathfrak{k}}(\\mathfrak{p})^c = \\{0\\}$. It then follows that $[E_{\\phi_j},\\mathfrak{p}^c]\\neq \\{0\\}$, hence there exists $\\gamma\\in R_\\mathfrak{p}$ with $[E_{\\phi_j},E_\\gamma]\\neq 0$, i.e. $\\phi_j+\\gamma \\in R_\\mathfrak{p}$. Now, if $\\gamma>0$, then $\\a:= \\phi_j+\\gamma\\in R_\\mathfrak{p}^+\\setminus\\Pi$ and $n_j(\\a)\\geq 1$. Suppose now $\\gamma<0$. We write $\\gamma=c_j\\phi_j + \\sum_{\\theta\\in \\Pi\\setminus{\\phi_j}} c_\\theta\\theta$ for some nonpositive integers $c_j,c_\\theta$. As $\\gamma\\neq -\\phi_j$, there exists at least one negative coefficient $c_\\theta<0$, for some $\\theta\\in \\Pi,\\theta\\neq\\phi_j$. Therefore the root $\\gamma+\\phi_j$ must be negative and $1+c_j\\leq 0$, i.e. $\\a:=-\\gamma\\in R_\\mathfrak{p}^+\\setminus\\Pi$ and $n_j(\\a)=-c_j\\geq 1$.\n\\end{proof}\n\n\nWe now fix a system of simple roots $\\Pi$ as in Lemma \\ref{L1}. In order to solve the corresponding system of equations \\eqref{sys1} for the positive unknowns $\\{g_i,h_j,g_\\a\\}$, we will show how to choose the positive values $\\{g_\\a\\}_{\\a\\in R^+\\setminus\\Pi}$ in such a way to guarantee that the constants $\\{g_i,h_j\\}$,\ndefined to satisfy \\eqref{sys1}, are positive.\\par \nWe set \n$$\\Sigma_\\mathfrak{k} := \\{\\a\\in R_\\mathfrak{k}^+|\\ \\a\\not\\in \\Pi, \\a\\in {\\rm{Span}}\\{\\Pi_{nc}\\}\\},\\qquad A_\\mathfrak{k} = (R_\\mathfrak{k}\\setminus \\Pi_c) \\setminus \\Sigma_\\mathfrak{k}.$$\nThen the system of equations \\eqref{sys1} can be written as \n\\begin{equation}\\label{sys2}\n\\left\\{\\begin{aligned}\ng_j &= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(\\a) - \\sum_{\\a\\in A_\\mathfrak{k}} g_\\a n_j(a),{}\\ \\  &j=1,\\ldots,k,\\ &(1)\\\\\nh_j &= \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(\\a) - \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),{}\\ \\  &j=1,\\ldots,l.\\ &(2)\n\\end{aligned}\\right.\\end{equation}\nWe start assigning $g_\\a=1$ for every $\\a\\in A_\\mathfrak{k}$. \\par \nThen, for every $j=1,\\ldots,k$, we use Lemma \\ref{L2} selecting a root $\\a\\in R_\\mathfrak{p}^+$ with $n_j(\\a)\\neq 0$, $\\a\\not\\in \\Pi$. This root $\\a$, which depends on $j$, contributes to the first sum in the righthandside of equation (1) in \\eqref{sys2} and the value $g_\\a$ can be chosen big enough so that $g_j$ is strictly positive. Summing up, we can assign values \n$\\{g_\\a\\}_{\\a\\in R_\\mathfrak{p}^+\\setminus \\Pi_{nc}}$ so that all $g_j$, $j=1,\\ldots,k$ can be defined as in \\eqref{sys2}, (1), and are strictly positive. \\par \nWe now turn to equation \\eqref{sys2}-(2), which can now be written as \n\\begin{equation}\\label{2}h_j= \\sum_{\\a\\in \\Sigma_\\mathfrak{k}} g_\\a m_j(\\a) +\\sum_{\\a\\in A_\\mathfrak{k}} m_j(\\a)  - \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),\\end{equation}\nwhere in the righthandside the last two sums have a fixed value. Now, by Lemma \\ref{L1}, we know that for every $j=1,\\ldots,l$, we can find $\\a\\in \\Sigma_\\mathfrak{k}$ with $m_j(\\a)\\neq 0$. These roots can be used to choose the coefficients $g_\\a$ big enough to guarantee that $h_j>0$, when defined to satisfy \\eqref{2}, is strictly positive.\\par \nIn order to complete the proof of our main result Theorem \\eqref{main}, we are left with the case $(\\gg_o,\\mathfrak{k})= (\\mathfrak{so}(1,2n),\\mathfrak{so}(2n))$ with standard system of simple roots $\\Pi=\\{\\epsilon_i-\\epsilon_{i+1},\\epsilon_n,\\ i=1,\\ldots,n-1\\}$, $\\Pi_{nc}=\\{\\epsilon_n\\}$. We see that \n$$R_\\mathfrak{k}^+=\\{\\epsilon_i\\pm\\epsilon_j,\\ i< j\\},\\quad R_\\mathfrak{p} =\\{\\epsilon_1,\\ldots,\\epsilon_n\\}.$$\nNow, we use equation \\eqref{eq} and search for positive real numbers $\\{x,y,z_i,\\ i=1,\\ldots,n\\}$ so that \n$$x\\cdot\\sum_{i<y}\\epsilon_i-\\epsilon_j + y\\cdot \\sum_{i< j} \\epsilon_i+e_j = \\sum_{i=1}^n z_i\\epsilon_i,$$\ni.e.\n$$\\sum_{i=1}^n [(x+y)(n-i)+(x-y)(i-1)]\\epsilon_i = \\sum_{i=1}^n z_i\\epsilon_i.$$\nIt is clear that the above equation has positive solutions by simply choosing $x>y>0$.\n\\begin{remark} We can consider the metric $h_o$ which coincides with $-B$ on the compact part $\\mathfrak{k}$, with $B$ on $\\mathfrak{p}$ and such that $h_o(\\mathfrak{k},\\mathfrak{p})=0$. This metric is easily seen to depend only on $\\gg_o$ and {\\it not} on the Cartan decomposition $\\gg_o=\\mathfrak{k}+\\mathfrak{p}$. We could then ask whether there exists a suitable complex structure such that the metric $h_o$ turns out to be balanced. The resulting equation has been already treated in \\cite{AP} and has a solution if and only if $\\gg_o = \\mathfrak{su}(p,p+1)\\cong \\mathfrak{su}(p+1,p)$ for $p\\geq 1$.   \n \\end{remark}\n\\section{Non-existence of pluriclosed metrics}\nIn this section we prove the following non-existence result\n\\begin{proposition} The compact complex manifolds ($\\rm{M},\\rm{J}$), wheer $\\rm{M}=\\rm{G}_o\/\\Gamma$, do not admit any pluriclosed metric.\n \\end{proposition}\nNote that in the above statement $\\rm{J}$ is the complex structure we have exhibited in section \\ref{proof}. \\par \nNow, if $h$ is any such metric, we can obtain a pluriclosed invariant metric $h$ on $\\rm{G}_o$ which is also invariant under right $\\T$-translations. It follows that on $\\gg$ we have \n$$h(\\gg_\\a,\\gg_\\b) = 0 \\quad {\\rm{if}}\\quad \\b\\neq-\\a.$$\nIn order to write down the condition $dd^c\\o=0$, where $\\o$ is the fundamental form of $h$, we recall the Koszul's formula for the differential of invariant forms. If $\\phi$ is any invariant $k$-form on $\\rm{G}_o$ or equivalently on $\\gg_o$, then for every $v_o,\\ldots,v_{k}$ in $\\gg_o$ \n$$d\\phi(v_o,\\ldots,v_{k}) = \\sum_{i<j}(-1)^{i+j}\\phi([x_i,x_j],\nv_1,\\ldots,\\widehat{v_i},\\ldots,\\widehat{v_j}\\ldots,v_{k}).$$\nWe set $\\phi:=d^c\\o$ and compute $d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b})$ for $\\a,\\b\\in R^+$. We have \n$$d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -\\phi(H_\\a,E_\\b,E_{-\\b}) +\n\\phi(N_{\\a\\b}E_{\\a+\\b},E_{-\\a},E_{-\\b})$$\n$$-\\phi(N_{\\a,-\\b}E_{\\a-\\b},E_{-\\a},E_\\b)-\\phi(N_{-\\a,\\b}E_{\\b-\\a},E_\\a,E_{-\\b})+\\phi(N_{-\\a,-\\b}E_{-\\a-\\b}E_\\a,E_\\b)$$\n$$- \\phi(H_\\b,E_\\a,E_{-\\a}), $$\nwhere we use the standard notation $[E_{\\gamma},E_{\\epsilon}]=N_{\\gamma,\\epsilon}E_{\\gamma+\\epsilon}$ for every $\\gamma,\\epsilon\\in R$. Using the known identities for the Weyl basis (see \\cite{He}, p. 172,176), we can write that \n$$d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -\\phi(H_\\a,E_\\b,E_{-\\b}) \n- \\phi(H_\\b,E_\\a,E_{-\\a})$$\n$$+2\n\\phi(N_{\\a\\b}E_{\\a+\\b},E_{-\\a},E_{-\\b}) - 2\\phi(N_{\\a,-\\b}E_{\\a-\\b},E_{-\\a},E_\\b).$$\nWe also introduce the notation $JE_\\gamma= i\\epsilon_\\gamma E_\\gamma$ for every $\\gamma\\in R$, where $\\epsilon_\\gamma=\\pm 1$ according to $\\gamma\\in R^{\\pm}$. Then\n$$dd^c\\o(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) =  -d\\o(JH_\\a,E_\\b,E_{-\\b}) -  d\\o(JH_\\b,E_\\a,E_{-\\a}) $$\n$$- 2iN_{\\a,\\b} d\\o(E_{\\a+\\b},E_{-\\a},E_{-\\b}) - 2iN_{\\a,-\\b}\\epsilon_{\\a-\\b}d\\o(E_{\\a-\\b},E_{-\\a},E_\\b).$$ \nNow we easily compute \n$$d\\o(JH_\\a,E_\\b,E_{-\\b}) = -\\o(H_\\b,JH_\\a)$$\nand \n$$d\\o(E_{\\a+\\b},E_{-\\a},E_{-\\b}) = N_{\\a,\\b}(\\o(E_\\a,E_{-\\a})+\\o(E_\\b,E_{-\\b}) - \\o(E_{\\a+\\b},E_{-\\a-\\b})),$$\nwhere we have used the fact that $N_{\\a,\\b}= N_{\\a+\\b,-\\b}=-N_{\\a+\\b,-\\a}$ (see \\cite{He}, p. 172). Similarly,\n$$d\\o(E_{\\a-\\b},E_{-\\a},E_\\b) = N_{\\a,-\\b}( - \\o(E_\\b,E_{-\\b}) + \\o(E_\\a,E_{-\\a})-\\o(E_{\\a-\\b},E_{\\b-\\a})).$$\nSumming up we have \n$$dd^c\\o(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -2\\o(JH_\\a,H_\\b)$$\n$$-2iN_{\\a,\\b}^2(\\o(E_\\a,E_{-\\a})+\\o(E_\\b,E_{-\\b}) - \\o(E_{\\a+\\b},E_{-\\a-\\b}))$$\n$$-2iN_{\\a,-\\b}^2\\epsilon_{\\a-\\b}(- \\o(E_\\b,E_{-\\b}) + \\o(E_\\a,E_{-\\a})-\\o(E_{\\a-\\b},E_{\\b-\\a})).$$\nWe now set $a_\\a : h(E_\\a,E_{-\\a})$. The pluriclosed condition implies that \n$$0=-h(H_\\a,H_\\b) -iN_{\\a,\\b}^2(-i a_\\a -i a_\\b + ia_{\\a+\\b})$$\n$$- iN_{\\a,-\\b}^2\\epsilon_{\\a-\\b}( -i\\epsilon_{\\b-\\a}a_{\\a-\\b}  + i a_\\b- ia_\\a)$$\nhence\n\\begin{equation}\\label{eq1} h(H_\\a,H_\\b) = N_{\\a,\\b}^2(a_{\\a+\\b}- a_\\a - a_\\b)\n+ N_{\\a,-\\b}^2\\epsilon_{\\a-\\b}( \\epsilon_{\\a-\\b}a_{\\a-\\b}  +  a_\\b-  a_\\a)\\end{equation}\nWe recall that \n$$a_\\a = h(E_\\a,E_{-\\a}) = - h(v_\\a,v_\\a) < 0,\\qquad \\a\\in R_\\mathfrak{k}^+,$$\n$$a_\\a = h(E_\\a,E_{-\\a}) = h(v_\\a,v_\\a) > 0, \\qquad \\a\\in R_\\mathfrak{p}^+,$$\n$$h(H_\\a,H_\\b) = -h(iH_\\a,iH_\\b)\\in \\mathbb R,\\qquad \nh(H_\\a,H_\\a) < 0.$$\n\nNow, we recall that the existence of the complex structure $\\rm{J}$, which we constructed in section \\ref{proof}, relies on Lemma \\ref{L1}. In particular, when $\\gg_o\\neq \\mathfrak{so}(1,2n)$, we have the existence of two simple roots $\\psi_1,\\psi_2\\in \\Pi_{nc}$ with $\\psi_1+\\psi_2=\\phi\\in R_\\mathfrak{k}$. The following lemma is elementary.\n\\begin{lemma} Either $\\psi_1+2\\psi_2\\not\\in R$ or $\\psi_2+2\\psi_1\\not\\in R$.\\end{lemma}\n\\begin{proof} As $\\psi_1,\\psi_2$ are simple, we have $\\pm(\\psi_1-\\psi_2)\\not\\in R$. Now, $\\psi_i+n\\psi_j\\in R$ if and only if $0\\leq n\\leq q_j$ with \n$q_j = -2\\frac{\\langle \\psi_1,\\psi_2\\rangle}{||\\psi_j||^2}\\in\\mathbb N$ for $i\\neq j$. It is then clear that $q_1,q_2\\geq 2$ is impossible, as $\\psi_1\\neq \\psi_2$ implies $q_1\\cdot q_2 < 4$.\\end{proof}\nSuppone then that $\\phi+\\psi_1 = \\psi_2+2\\psi_1\\not \\in R$. We now apply \\eqref{eq1} with two possible choices for $\\a,\\b$, namely:\\par \\medskip\n\\noindent (1)\\ $\\a=\\psi_1,\\b=\\psi_2$. Then \n$$h(H_{\\psi_1},H_{\\psi_2}) = N_{\\psi_1,\\psi_2}^2(a_\\phi-a_{\\psi_1}-a_{\\psi_2}).$$\n(2)\\ $\\a=\\phi,\\b=\\psi_1$. Then \n$$h(H_{\\phi},H_{\\psi_2}) = N_{\\phi,-\\psi_1}^2(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi}).$$\nSubtracting (1) from (2) we get \n$$h(H_{\\psi_2},H_{\\psi_2}) = \\left(N_{\\phi,-\\psi_1}^2+N_{\\psi_1,\\psi_2}^2\\right)(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi}).$$\nThis is a contradiction, as $h(H_{\\psi_2},H_{\\psi_2})<0$, while \n$a_{\\psi_i}>0$ for $i=1,2$ and $a_\\phi<0$.\\par \nWe are left with the case $\\gg_o=\\mathfrak{so}(1,2n)$, that we have dealt with separately in section \\ref{proof}. In this case the complex structure $\\rm{J}$ is defined by the standard system of positive roots, namely $R^+=\\{\\epsilon_i\\pm\\epsilon_j, \\epsilon_i,\\ 1\\leq i\\neq j\\leq n\\}$. In particular $R_\\mathfrak{k}^+ =\\{\\epsilon_i\\pm\\epsilon_j\\}_{i\\neq j}$ and $R_\\mathfrak{p}^+=\\{\\epsilon_i\\}_{i=1,\\ldots,n}$. We now consider $\\psi_i=\\epsilon_i$, $i=1,2$, $\\phi_1=\\psi_1+\\psi_2\\in R_\\mathfrak{k}^+$ and $\\phi_2=\\psi_1-\\psi_2\\in R_\\mathfrak{k}^+$.   We apply \\eqref{eq1} in two different ways: \\par\n\\noindent (1)\\ $\\a=\\psi_1,\\b=\\psi_2$. Then \n$$h(H_{\\psi_1},H_{\\psi_2}) = N_{\\psi_1,\\psi_2}^2(a_\\phi-a_{\\psi_1}-a_{\\psi_2}) + N_{\\psi_1,-\\psi_2}^2(a_{\\phi_2}+a_{\\psi_2}-a_{\\psi_1}).$$\n(2)\\ $\\a=\\phi_1,\\b=\\psi_2$. Note that $\\phi_1+\\psi_1\\not\\in R$. Then \n$$h(H_{\\phi_1},H_{\\psi_1}) =  N_{\\phi_1,-\\psi_1}^2(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi_1}).$$\nTherefore \n$$h(H_{\\psi_1},H_{\\psi_1}) = (N_{\\phi_1,-\\psi_1}^2+N_{\\psi_1,\\psi_2}^2)(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi_1})+ N_{\\psi_1,-\\psi_2}^2(a_{\\psi_1} -a_{\\phi_2}-a_{\\psi_2})$$\nWe now recall that, if $\\gamma,\\d\\in R$, then $N_{\\gamma,\\d}^2= \\frac{q(1-p)}{2}||\\gamma||^2$, where $\\d+n\\gamma$, $p\\leq n\\leq q$, is the $\\gamma$-series containing $\\d$ (see \\cite{He}, p.176). We then immediately see that $N_{\\psi_1,\\psi_2}^2 = N_{\\psi_1,-\\psi_2}^2$ and noting furthermore that $N_{\\phi_1,-\\psi_1}^2 = N_{\\psi_1,\\psi_2}^2$,  we can write \n$$h(H_{\\psi_1},H_{\\psi_1}) = N_{\\psi_1,\\psi_2}^2(a_{\\psi_2}+3 a_{\\psi_1}-2a_{\\phi_1}- a_{\\phi_2}),    $$\ngiving the contradiction $h(H_{\\psi_1},H_{\\psi_1}) >0$.\n\n\n\n\\section{Geometric properties}\nIn this section, we prove the following result, which may contribute to shed to some light on the geometry of the complex balanced manifolds we have constructed in the previous sections. \n\\begin{theorem} If $(\\rm{M},\\rm{J},h)$ is a balanced $n$-dimensional manifold, where $\\rm{M}=\\rm{G}_o\/\\Gamma$, $\\rm{J}$ is a standard invariant complex structure and $h$ is a balanced Hermitian metric, then the metric $h$ has vanishing Chern scalar curvature.\\par \nMoreover $c_1(\\rm{M})=0$ and the Kodaira dimension $\\kappa(M) = -\\infty$.\n \\end{theorem}\n\nWe consider a standard complex structure $\\rm{J}$ on a manifold $\\rm{M} = \\rm{G}_o\/\\Gamma$. We denote by $D$ the Chern connection relative to a Hermitian metric $h$ which is induced by an invariant metric on $\\rm{G}_o$, again denoted by $h$. We can moreover suppose that $h$ is invariant by the right $\\T$-translations. \\par \nIf $x\\in\\gg_o$ and if we still denote by $x$ the induced left-invariant vector field on $\\rm{G}_o$, we consider $D_x\\in\\End(\\gg_o)$ the endomorphism of $\\gg_o$ which assigns to every $y\\in\\gg_o$ the element $D_xy$ corresponding to the left invariant vector field $D_xy$. Clearly $D_x\\in \\mathfrak{so}(\\gg_o,h)$ and $[D_x,\\rm{J}]=0$. Moreover \n\\begin{equation}\\label{chern}D_xy = [x,y]^{10},\\qquad \\forall x\\in \\gg_o^{01},\\ y\\in \\gg_o^{10},\\end{equation}\nthat follows from the fact that $T^{1,1}=0$, where $T$ is torsion of $D$.\\par \nIf $R$ denote the curvature, where $R_{xy} = [D_x,D_y]- D_{[x,y]}$, we are interested in the first Ricci tensor $\\rho$ given by\n$$\\rho(x,y) = -\\frac 12{\\rm{Tr}}(J\\circ R_{xy}).$$\nAs the complex structure and the metric are both invariant under the adjoint action of the group $T = \\exp(\\mathfrak{t})$, we see that \n$$\\rho(\\mathfrak{t},E_\\a) = 0,\\ \\forall \\a\\in R,$$\n$$\\rho(E_\\a,E_\\b)\\neq 0\\ {\\rm{implies}}\\ \\b=-\\a,\\ \\a,\\b\\in R.$$\nTherefore we can compute \n$$\\rho(E_\\a,E_{-\\a}) = \\frac 12 {\\rm{Tr}}(JD_{H_\\a}).$$\n\\begin{lemma} For every $x\\in \\mathfrak{h}$\n$$D_x = \\ad(x).$$\n\\end{lemma}\n\\begin{proof} We use similar arguments as in \\cite{Po}. It will suffice to consider the case where $x\\in \\mathfrak{h}^{10}$; then for every $\\a\\in R^+$ we have \n$$D_xE_{-\\a} = [x,E_{-\\a}]^{01} = [x,E_{-\\a}],\\quad D_x\\mathfrak{h}^{01} = 0$$\nby \\eqref{chern}. Then if $\\b\\in R^+$ we have \n$$h(D_xE_\\a,E_{-\\b}) = - h(E_\\a,D_xE_{-\\b}) = -\\b(x)h(E_\\a,E_{-\\b}) = 0 \\qquad {\\rm{if}}\\ \\a\\neq\\b,$$\nso that $D_xE_\\a=\\a(x)E_\\a = [x,E_\\a]\\ ({\\rm{mod}}\\ \\mathfrak{h})$. As $h(D_xE_\\a,\\mathfrak{h}^{01}) = -h(E_\\a,D_x\\mathfrak{h}^{01}) = 0$, we conclude that \n$$D_xE_\\a = [x,E_\\a].$$\nFinally, $h(D_x\\mathfrak{h}^{10},\\mathfrak{h}^{01})=0$ and $h(D_x\\mathfrak{h}^{10},E_{-\\a}) = \n-h(\\mathfrak{h}^{10},[x,E_{-\\a}]) = 0$, so that $D_x\\mathfrak{h} = 0 = [x,\\mathfrak{h}]$.\\end{proof}\n\nIt follows that \n$$\\rho(\\mathfrak{h},\\mathfrak{h})=0$$ \nand \n$$\\rho(E_\\a,E_{-\\a}) = \\frac 12\\left(2\\sum_{\\beta\\in R^+}i\\beta(H_\\a)\\right) = B(H_\\a,\\delta),$$\nwhere \n$$\\delta = \\sum_{\\beta\\in R^+} iH_\\b\\in \\mathfrak{t} .$$\nThis means that for every $\\a,\\b\\in R$ we have\n$$\\rho(E_\\a,E_\\b) = - B(E_\\a,[\\d,E_\\b]) = B([E_\\a,E_\\b],\\d)$$\nand therefore for every $x,y\\in \\gg_o$\n\\begin{equation} \\label{rho}\\rho(x,y) = B([x,y],\\delta).\\end{equation}\nThis means that $\\rho = d\\phi$, where $\\phi$ is the left-invariant $1$-form that is given by $\\phi(v) = B(v,\\delta)$. Then clearly $c_1(M)=0$.\\par\nWe now show that the tensor powers $K_{\\rm{M}}^{\\otimes k}$ are holomorphically non trivial for every $m\\geq 1$. Indeed, the metric $h$ induces a Hermitian metric on the line bundles $K_{\\rm{M}}^{\\otimes m}$ with curvature form $m\\rho$. If $\\Omega$ is a nowhere vanishing holomorphic section of $K_{\\rm{M}}^{\\otimes k}$ , then $m\\rho =  \n-i\\partial\\overline\\partial \\ln(||\\Omega||^2)$. \nIf we denote by\\, $\\widehat{}$\\,  the result of the symmetrization process, which commutes with the operators $\\partial$ and $\\overline\\partial$, we obtain on $\\rm{G}_o$ that $\\widehat{\\rho} =-i \\partial\\overline\\partial\\widehat{\\ln(||\\Omega||^2)} = 0$. As $\\rho$ is invariant, $\\widehat\\rho=\\rho=0$ and we get a contradiction as $\\d\\ne 0$.\\par \nWe now compute the Chern scalar curvature $s^{Ch}$ of the metric $h$ using formula \\eqref{rho}. We use the orthonormal frame $e_1,\\ldots,e_{2n}$. Then \n$$s^{Ch} = \\sum_i \\rho(Je_i,e_i) = \n-2\\sum_{\\a\\in R^+} \\frac{1}{h_\\a^2} \\rho(v_\\a,w_\\a) = $$\n$$= 2iB\\left(\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac{1}{h_\\a^2} H_\\a-\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac{1}{h_\\a^2} H_\\a,\\d\\right) = 0$$\nif we consider the system of positive roots satisfying equation \\eqref{eq}.\nThe claim $\\kappa(\\rm{M})=-\\infty$ now follows from \nThm. 1.4 in \\cite{Y}.\\par \n\n\n\n\n\n\\begin{remark} Note that also for a compact group $\\K$ endowed with an invariant complex structure we have $h^{n,0}(\\K)=0$, see \\cite{Pi}, Prop. 3.7.\\par\nWe finally remark here that the balanced condition implies that the two scalar curvatures that one can obtain tracing the Chern curvature tensor coincide (see \\cite{Ga3}, p. 501).\\end{remark}\n\n\n\n\n\n\n\n\n\n\\vspace{2cm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}