Nicholas Proudfoot and Benjamin Young Department of Mathematics, University of Oregon, Eugene, OR 97403

Transcription

1 Confguraton spaces, FS op -modules, and Kazhdan-Lusztg polynomals of brad matrods Ncholas Proudfoot and Benjamn Young Department of Mathematcs, Unversty of Oregon, Eugene, OR Abstract. The equvarant Kazhdan-Lusztg polynomal of a brad matrod may be nterpreted as the ntersecton cohomology of a certan partal compactfcaton of the confguraton space of n dstnct labeled ponts n C, regarded as a graded representaton of the symmetrc group S n. We show that, n fxed cohomologcal degree, ths sequence of representatons of symmetrc groups naturally admts the structure of an FS-module, and that the dual FS op -module s fntely generated. Usng the work of Sam and Snowden, we gve an asymptotc formula for the dmensons of these representatons and obtan restrctons on whch rreducble representatons can appear n ther decomposton. 1 Introducton Gven a matrod M, the Kazhdan-Lusztg polynomal P M t) was defned n [EPW16]. More generally, f M s equpped wth an acton of a fnte group W, one can defne the W -equvarant Kazhdan-Lusztg polynomal PM W t) [GPY17]. By defnton, P M W t) s a graded vrtual representaton of W, and takng dmenson recovers the non-equvarant polynomal. These representatons have been computed when M s a unform matrod [GPY17, Theorem 3.1] and conjecturally for certan graphcal matrods [Ged, Conjecture 4.1]. However, n the case of the brad matrod the matrod assocated wth the complete graph on n vertces), very lttle s known. The non-equvarant verson of ths problem was taken up n [EPW16, Secton 2.5] and the S n -equvarant verson n [GPY17, Secton 4], but wth few concrete results or even conjectures. In ths paper we use an nterpretaton of the equvarant Kazhdan-Lusztg polynomal of the brad matrod M n as the ntersecton cohomology of a certan partally compactfed confguraton space to show that, n fxed cohomologcal degree, t admts the structure of an FS-module, as studed n [Pr00, CEF15, SS17]. Applyng the results of Sam and Snowden [SS17], we use the FSmodule structure or, more precsely, the dual FS op -module structure) to mprove our understandng of ths sequence of representatons. In partcular, we obtan the followng results Corollary 6.2): For fxed, we prove that the generatng functon for the th non-equvarant Kazhdan-Lusztg coeffcent of M n wth n varyng) s a ratonal functon wth poles lyng n a prescrbed set. For fxed, we derve an asymptotc formula for the th non-equvarant Kazhdan-Lusztg coeffcent of M n n terms of another Kazhdan-Lusztg coeffcent that depends only on. We show that, f λ s a partton of n and the assocated Specht module V λ appears as a summand of the th equvarant Kazhdan-Lusztg coeffcent of M n, then λ has at most 2 rows. 1

2 We also produce relatve versons of these results n whch we start wth an arbtrary graph Γ and consder the sequence of graphs whose n th element s obtaned from Γ by addng n new vertces and connectng them to everythng ncludng each other). The orgnal problem s the specal case where Γ s the empty graph. Acknowledgments: The authors are grateful to Steven Sam and John Wltshre-Gordon for extremely helpful dscussons wthout whch ths paper would not have been wrtten, and to Tom Braden for greatly clarfyng the materal n Secton 3. The frst author s supported by NSF grant DMS Kazhdan-Lusztg polynomals and confguraton spaces Let M be a matrod on the ground set I, equpped wth an acton of a fnte group W. Ths means that W acts on I by permutatons and that the acton of W takes bases to bases. An equvarant realzaton of W M s W -subrepresentaton V C I such that B I s a bass for M f and only f V projects somorphcally onto C B. Note that we have C I CP 1) I, sttng nsde as the locus of ponts wth no coordnate equal to. More generally, for any subset S I, let p S CP 1) I be the pont wth ps ) = 0 for all S and p S ) j = for all j S c, and let C I S := {p CP 1) I p for all S and p 0 for all S c} be the standard affne neborhood of p S. Thus p I = 0 V C I = C I I. Gven a W -subrepresentaton V C I, we defne the followng three spaces wth W -actons: UV ) := V C ) I, the complement of the coordnate hyperplane arrangement n V, Y V ) := V CP 1) I, the Schubert varety of V see [AB16] or [PXY, Secton 7]), XV ) := Y V ) C I, the recprocal plane of V. Note that Y V ) s a compactfcaton of UV ), whle V and XV ) are each partal compactfcatons of UV ). Let CM, W denote the coeffcent of t n the equvarant Kazhdan-Lusztg polynomal PM W t) of W M. The followng theorem appears n [GPY17, Corollary 2.12] as an applcaton of the work n [PWY16, Secton 3]. Theorem 2.1. If V C I s an equvarant realzaton of W M, then CM, W s somorphc as a representaton of W to the ntersecton cohomology group IH 2 XV ); C ). In partcular, CM, W s an honest not just vrtual) representaton. Let I n := {, j) j [n] }, and let M n be the matrod on the ground set I n whose bases consst of orented spannng trees for the complete graph on n vertces. We wll refer to M n as the brad matrod, whch comes equpped wth a natural acton of the symmetrc group S n. 2

3 Remark 2.2. It s more standard to defne the brad matrod on the ground set of unordered pars of elements of [n]. Our matrod M n s not smple for any j, the set {, j), j, )} s dependent), and ts smplfcaton s S n -equvarantly somorphc to the usual brad matrod. In partcular, they have the same lattce of flats see Secton 3 for the defnton of a flat), and therefore the same equvarant Kazhdan-Lusztg polynomal. We prefer the ordered verson because t s equvarantly realzable as we explan below), thus we may apply Theorem 2.1. Consder the lnear map f : C n C In gven by f j z 1,..., z n ) = z z j. The kernel of f s equal to the dagonal lne C C n, so f descends to an ncluson of V n := C n /C nto C In, whch gves an equvarant realzaton of C n. Let U n := UV n ), Y n := Y V n ), and X n := XV n ). The space U n may be dentfed wth the confguraton space of n dstnct labeled ponts n C, modulo smultaneous translaton. Informally, V n s obtaned from U n by allowng the dstances between ponts to go to zero, the recprocal plane X n s obtaned from U n by allowng the dstances between ponts to go to nfnty, and the Schubert varety Y n s obtaned from U n by allowng dstances between ponts to go to ether zero or nfnty. Remark [ 2.3. ] The recprocal plane X n may also be descrbed as the spectrum of the subrng C j of the rng of ratonal functons on C n. More generally, XV ) s somorphc to the 1 x x j spectrum of the subrng of ratonal functons on V generated by the recprocals of the coordnate functons. Ths rng s called the Orlk-Terao algebra of V C I. The non-equvarant Kazhdan-Lusztg polynomal of M n for n 20 appears n [EPW16, Secton A.2]. The frst few coeffcents of ths polynomal can be expressed n terms of Strlng numbers [EPW16, Corollary 2.24 and Proposton 2.26]. The same can be sad of all of the terms, but the expressons become ncreasngly complcated. Indeed, the th coeffcent can be expressed as an alternatng sum of -fold products of Strlng numbers, where the number of summands s equal to [PXY, Corollary 4.5]. We also made a conjecture about the leadng term when n s even [EPW16, Secton A]. The degree of the Kazhdan-Lusztg polynomal s by defnton strctly less than half of the rank of the matrod, so the largest possble degree of P M2 t) s 1. Conjecture 2.4. For all > 0, C M2, 1 = 2 3)!!2 1) 2, the number of labeled trangular cact on 2 1) nodes [Slo14, Sequence A034941]. The equvarant Kazhdan-Lusztg polynomal of the brad matrod s even more dffcult to understand. The lnear term s computed n [GPY17, Proposton 4.4], and we also compute the remanng coeffcents for n 9 [GPY17, Secton 4.3]. We also gve a functonal equaton that characterzes the generatng functon for the Frobenus characterstcs of the equvarant Kazhdan- Lusztg polynomals [GPY17, Equaton 7)], but we do not know how to solve ths equaton. 3 The spectral sequence In ths secton we explan how to construct a spectral sequence to compute the ntersecton cohomology of the recprocal plane, whch we wll later use to endow the Kazhdan-Lusztg coeffcents 3

4 of brad matrods wth an FS-module structure. Ths constructon appears for a partcular example n [PWY16, Secton 3], and we make some remarks there about how to generalze the constructon to arbtrary V C I. We wll gve the constructon n full generalty here, takng care to emphasze the functoralty, whch wll be crucal for our applcaton n Secton 6. A subset F I s called a flat of M f there exsts a pont v V such that F = { v = 0}. Gven a flat F, let V F := V C F c C F c and let V F C F be the mage of V along the projecton C I C F. The dmenson of V F s called the rank of F, whle the dmenson of V F s called the corank. Gven a flat F I, let Y V ) F := { p Y V ) p = F c}. Then we have Y V ) = F Y V ) F 1) and Y V ) F = VF for all F [PXY, Lemmas 7.5 and 7.6]. Ths affne pavng may also be descrbed as the orbts of a group acton. The addtve group C acts on CP 1 = C { } by translatons; takng products, we obtan an acton of C I on CP 1) I. The subgroup V C I acts on the subvarety Y V ) := V CP 1) I, and the subset Y V )F s equal to the orbt of the pont p F Y V ). The stablzer of p F s equal to V F V, and the orbt s therefore somorphc to V/V F = V F. For any flat F I, there s a canoncal ncluson ɛ F : XV F ) Y V ) C I F by the formula In partcular, ɛ F ) = p F. Consder the map p f F c ɛ F p) := 0 f F. ϕ F : V XV F ) Y V ) v, p) v ɛ F p). defned explctly If we choose a secton s : V F V of the projecton π F : V V F, then the restrcton of ϕ F to sv F ) XV F ) s an open mmerson. In partcular, for every v V, the map ϕ F,v : XV F ) Y V ) takng p to ϕ F v, p) s a normal slce to the stratum V F Y V ) at the pont ϕ F,v ) = π F v) V F. Intersectng the stratfcaton n Equaton 1) wth C I, we obtan a stratfcaton XV ) = F UV F ) of the recprocal plane XV ), whch can be used to construct a spectral sequence that computes the ntersecton cohomology of XV ). Theorem 3.1. Let W be a fnte group actng on I, and let V C I be a W -subrepresentaton. There exsts a frst quadrant cohomologcal spectral sequence EV, ) n the category of W -representatons 4

5 wth EV, ) p,q 1 = crk F =p H 2 p q UV F ); C ) IH 2 q) XV F ); C ), convergng to IH 2 XV ); C). Proof. Let ι F : V F Y V ) denote the ncluson of the stratum of Y V ) ndexed by F, whch restrcts to the ncluson ι F : UV F ) XV ) of the correspondng stratum of XV ). The stratfcaton of XV ) nduces a fltraton by supports on the complex of global sectons of an njectve resoluton of the ntersecton cohomology sheaf IC XV ). Ths fltered complex gves rse to a spectral sequence EV ) wth EV ) p,q 1 = crk F =p convergng to IH XV ); C) [BGS96, Secton 3.4]. H p+q ι! F IC XV ) ) The sheaf ι! F IC XV ) s a pror a local system on UV F ) wth fbers equal to the compactly supported ntersecton cohomology of the stalks of IC XV ). However, snce XV ) s open n Y V ), the sheaf ι! F IC XV ) on UV F ) concdes wth the restrcton of the sheaf ι! F IC Y V ) on V F. Snce V F s a vector space, ths local system s trval. Even better, we have a canoncal trvalzaton. For any v F V F, we can choose v V wth π F v) = v F, and the slce ϕ F,v : XV F ) Y V ) nduces an somorphsm from the fber of ι! F IC Y V ) to the compactly supported ntersecton cohomology group IH c XV F ); C ). Snce the kernel V F of π F s connected, ths somorphsm does not depend on the choce of v. Thus we have a canoncal somorphsm EV ) p,q 1 = crk F =p j+k=p+q H j UV F ); C ) IH k c XV F ); C ). We now consder the weght fltraton on EV ), and pass to the maxmal subquotent EV, ) of weght 2. The group H j UV F ); C ) s pure of weght 2j [Sha93]; the groups IHc k XV F ); C ) and IH k XV ); C ) are both pure of weght k, and they vansh when k s odd [EPW16, Proposton 3.9]. Ths mples that EV, ) p,q 1 = crk F =p H 2j p q UV F ); C ) IHc 2p+q ) XV F ); C ), and that EV, ) converges to IH 2 XV ); C ). Fnally, we observe that dm XV F ) = crk F = p, so Poncaré dualty tells us that IHc 2p+q ) XV F ); C ) = IH 2 q) XV F ); C ). Remark 3.2. The proof of Theorem 3.1 for a partcular class of examples appears n [PWY16, Proposton 3.3]. The argument here s essentally the same. Indeed, we mplctly used Theorem 3.1 n the proof of Theorem 2.1, whch orgnally appeared n [GPY17, Corollary 2.12]. The only new ngredent here s an emphass of the fact that the local system ι! F IC XV ) s canoncally trvalzed, whch we need n order to make sense of Theorem 3.3. We are grateful to Tom Braden for explanng to us how ths works. 5

6 Next, we wll show that for every flat F I, we obtan a canoncal map from EV, ) to EV F, ), whch we wll descrbe explctly. The cohomology of UV ) s generated by degree 1 classes {ω I}. UV ) C I I. Theorem 3.3. Suppose that F I s a flat. Explctly, we have ω = [d log z ], where z s the coordnate functon on 1. There s a canoncal map of graded vector spaces IH XV ); C ) IH XV F ); C ), equvarant for the stablzer W F W of F. 2. There s a canoncal map of spectral sequences EV, ) EV F, ), equvarant for the stablzer W F W of F, convergng to the map n part If G F, then the compostons IH XV ); C ) IH XV F ); C ) IH XV G ); C ) and EV, ) EV F, ) EV G, ) concde wth the maps IH XV ); C ) IH XV G ); C ) and EV, ) EV G, ), respectvely. 4. The map from EV, ) p,q 1 = crk G=p H 2 p q UV G ); C ) IH 2 q) XV G ); C ) to EV F, ) p,q 1 = H 2 p q UVG F ); C ) IH 2 q) XV G ); C ) G F crk G=p klls summands wth G F. If G F and G, then the map on G summands s nduced by the map H 1 UV G ); C ) H 1 UV F G ); C) obtaned by settng ω equal to zero for all F. Proof. For any pont v F UV F ) V F, we have a map IH XV ); C ) H IC XV ),vf ) = H IC Y V ),vf ) = H IC XV F ), ) = IH XV F ); C ), where the second somorphsm s nduced by the slce ϕ F,v : XV F ) Y V ) for any v V such that π F v) = v F and the thrd somorphsm s nduced by the contractng acton of C on XV F ) [Spr84, Corollary 1]. As before, the fact that ths map s ndependent of the choce of v follows from the fact that the kernel V F of π F s connected. Snce the codmenson p strata of XV F ) concde wth the premages of the codmenson p strata of Y V ), the fltratons of IC Y V ),vf = ICXV F ), nduced by the two stratfcatons concde, thus ths map nduces a map of spectral sequences assocated wth the stratfcatons. Ths proves the frst two parts of the theorem. To prove the thrd part of the theorem, choose generc elements v, v V and v V F such that v = v + v. We then have maps ϕ G,v : XV G ) Y V ), ϕ F,v : XV F ) Y V ), and ϕ F G,v : XV G ) Y V F ). 6

7 If p XV G ) s suffcently close to the pont more precsely, f p > v for all Gc ), then ϕ F G,v p) XV F ). Thus the composton ϕ F,v ϕ F G,v s well defned n a neghborhood of XV G ), and on that neghborhood we have ϕ G,v = ϕ F,v ϕ F G,v. Snce the maps n parts 1 and 2 are determned by the behavor of the slce map n a neghborhood of, ths mples that the maps compose as desred. To prove the last part of the theorem, we need to understand explctly the map from the G stratum of XV F ) to the G stratum of Y V ). Specfcally, f p UVG F ), and G, then p + v f F c ϕ F,v p) = v f F. As n the prevous paragraph, f we restrct to the open set B UV F G ) on whch each p has norm larger than v, then our map wll take values n UV G ). Note that B s homotopy equvalent to UV F G ), and the map n the spectral sequence s determned by the pullback map from H UV G ); C) to H B; C) = H UV F G ); C). Let z be the th coordnate functon on UV G ), so that ω = [d log z ]. If F, then z pulls back to a constant functon, so ω pulls back to zero. If G F, then z pulls back to z v, so ω pulls back to [d logz v )] = [d logz 1 v /z ))] = [d log z ] + [d log1 v /z )] = ω + [d log1 v /z )]. Snce the norm of z s always greater than the norm of v on B, the real part of 1 v /z s always postve, whch mples that d log1 v /z ) s exact. Thus ω pulls back to ω, as desred. We now unpack Theorem 3.1 n the specal case where I = I n and V = V n. In ths case, flats are n bjecton wth set-theoretc parttons of [n]. More precsely, gven a partton of [n], the set of all ordered pars, j) such that and j le n the same block of the partton s a flat, and every flat arses n ths way. A flat of corank p corresponds to a partton nto p + 1 unlabeled) blocks P 1,..., P p+1. Gven such a flat F, we have UV n ) F ) = U P1 U Pp+1 and XV F n ) = X p+1. In order to clarfy the ssue of labeled versus unlabeled parttons, we make the followng defntons: A p,q n) := f:[n] [p+1] ) H 2 p q U f 1 1) U f 1 p+1) ; C IH 2 q) X p+1 ; C) and B p,q n) := A p,q n) S p+1, where S p+1 acts on [p + 1]. Thus we have the followng corollary of Theorem

8 Corollary 3.4. There exsts a frst quadrant cohomologcal spectral sequence En, ) n the category of S n -representatons wth En, ) p,q 1 = B p,q n) convergng to IH 2 X n ). Remark 3.5. The reason for usng homology rather than cohomology n the defnton of A p,q n) and then undong ths by dualzng n Corollary 3.4) wll become clear n Secton 6. Brefly, the explanaton s that ntersecton cohomology admts the structure of an FS-module and ntersecton homology admts the structure of an FS op -module, and t s the FS op -module structure that wll prove to be more useful. 4 FS-modules and FS op -modules Let FS be the category whose objects are nonempty fnte sets and whose morphsms are surjectve maps. An FS-module s a covarant functor from FS to the category of complex vector spaces, and an FS op -module s a contravarant functor from FS to the category of complex vector spaces. If N s an FS-module or an FS op -module, we wrte Nn) := N[n]), whch we regard as a representaton of the symmetrc group S n = Aut FS [n]). Let FA be the category whose objects are nonempty fnte sets and whose morphsms are all maps. For any postve nteger m, let P m := C{Hom FS, [m])} be the FS op -module that takes a fnte set E to the vector space wth bass gven by surjectons from E to [m]; ths s a projectve FS op -module called the prncpal projectve at m. We say that an FS op -module N s fntely generated f t s somorphc to the quotent of a fnte sum of prncpal projectves, and we say that t s fntely generated n degrees d f one only needs to use P m for m d. Ths s equvalent to the statement that, for any fnte set E and any vector v NE), we can wrte v as a fnte lnear combnaton of elements of the form f x), where f : E [m] and x Nm) for some m d. We call an FS op -module d-small f t s a subquotent of a module that s fntely generated n degrees d. A d-small FS op -module s always fntely generated [SS17, Corollary 8.1.3], but not necessarly n degrees d. For any partton λ = λ 1,..., λ lλ) ) n, let V λ be the correspondng rreducble representaton of S n. If λ s a partton of k and n k + λ 1, let λn) be the partton of n obtaned by addng a part of sze n k. For any FS op -module N, consder the ordnary generatng functon and the exponental generatng functon H N u) := G N u) := u n dm Nn), n=1 n=1 u n dm Nn). n! For any natural number d, let dm Nn) r d N) := lm n d n, 8

9 whch may or may not exst. The statements and proofs of the followng results were communcated to us by Steven Sam. Theorem 4.1. Let N be a d-small FS op -module. 1. If λ n and Hom Sn V λ, Nn)) 0, then lλ) d. 2. For any partton λ wth n λ + λ 1, dm Hom Sn Vλn), Nn) ) s bounded by a polynomal n n of degree at most d The ordnary generatng functon H N u) s a ratonal functon whose poles are contaned n the set {1/j 1 j d}. 4. There exsts polynomals p 0 u),..., p d u) such that the exponental generatng functon G N u) s equal to d p j u)e ju. j=0 5. The functon H N u) has at worst a smple pole at 1/d. Equvalently, the lmt r d N) exsts, and the polynomal p d u) n statement 4 s the constant functon wth value r d N). Proof. To prove statements 1 and 2, t s suffcent to prove them for the prncpal projectve P m for all m d. Let Q m ) := C{Hom FA, [m])}, so that P m s a submodule of Q m. Then Q m n) = C m ) n, and Schur-Weyl dualty tells us that the multplcty of V λ n ths representaton s equal to the dmenson of the representaton of GLm; C) ndexed by λ. In partcular, t s zero unless λ has at most m parts, and the dmenson of the representaton ndexed by λn) s bounded by a polynomal n n of degree at most m 1. Statements 1 and 2 follow for Q m, and therefore for P m. If N s fntely generated n degrees d, then statement 3 holds for N by [SS17, Corollary 8.1.4]. If N s a subquotent of N, then t s stll fntely generated n degrees r for some r, so statement 3 holds for N wth d replaced by r. But, snce N s a subquotent of N, we have dm Nn) dm N n) for all n, whch mples that e j = 0 for all j r. Statement 4 follows from statement 3 by fndng a partal fractons decomposton of the ordnary generatng functon, as observed n [SS17, Remark 8.1.5]. To prove statement 5, t s agan suffcent to consder P m for all m d. We have dm P m n) = Hom FS [n], [m]) Hom FA [n], [m]) = m n d n. Snce N s a subquotent of a fnte drect sum of modules of ths form, the dmenson of Nn) s bounded by a constant tmes d n. We now record a par of lemmas that say that certan natural constructons preserve smallness. 9

10 Lemma 4.2. Fx a natural number k, a k-tuple of natural numbers d 1,..., d k ), and a collecton of FS op -modules N 1,..., N k such that N s d -small. Let d = d d k. Then the FS op -module N gven by the formula s d-small. NE) := f:e [k] N 1 f 1 1)) N k f 1 k)) Proof. Snce d-smallness s preserved by takng drect sums and passng to subquotents, we may assume that N = P m NE) = = = for some m d. Then f:e [k] f:e [k] f:e [k] P m1 f 1 1)) P mk f 1 k)) C { Hom FS f 1 1), [m 1 ] )} C { Hom FS f 1 k), [m k ] )} { C Hom FS f 1 1), [m 1 ] ) Hom FS f 1 k), [m k ] )} = { C Hom FS E, [m1 ] [m k ] )} = { C Hom FS E, [m1 + + m k ] )} = P m1 + +m k E), so N s d-small. Lemma 4.3. Let N be d-small and let S be any set. Let N S be the FS-module defned by puttng N S E) := NS E) for all E, wth maps defned n the obvous way. Then N S s also d-small. Proof. As n the proof of Lemma 4.2, we may reduce to the case where N = P m for m d. In ths case, t s suffcent to show that every surjecton f : S E [m] factors as g d S h), where g s a surjecton from S [j] to [m] for some j m and h s a surjecton from [m] to [j]. It s clear that we can do ths by takng j to be the cardnalty of fe). Remark 4.4. The functor N N S s called a shft functor, and the analogous operaton for FI-modules has appeared n many contexts; see, for example, [CEFN14, Secton 2]. Fnally, the followng lemma wll be needed n the proof of Theorem 6.1. Lemma 4.5. Suppose that N N N s a complex of d-small FS op -modules, and let H denote ts homology n the mddle. If r d N) = 0 = r d N ), then r d H) = r d N ). Proof. Ths follows from the fact that dm N n) dm Nn) dm N n) dm Hn) dm Nn) and the defnton of r d. 10

11 5 Confguratons of ponts n the plane For any fnte set E, let ConfE) be the space of njectve maps from E to R 2. Arnol d [Arn69] proved that H ConfE); C) = Λ C [x j, j E] / x, x j x j, x j x jk + x jk x k + x k x j. Let H E) := H ConfE); C) and H E) := H ConfE); C) = H ConfE); C). Gven a map f : E F, we have a map H ConfE); C) H ConfF ); C) takng x j to x f)fj). Ths gves H the structure of an FA-module and H the structure of an FA op -module. Snce FS s a subcategory of FA, we may regard H as an FS-module and H as an FS op -module. Proposton 5.1. The FS op -module H 0 s 1-small. If 1, then H s 2-small and r 2 H ) = 0. Proof. We have H 0 = P1, whch s by defnton 1-small. Snce H E) s generated n degree 1, H E) s a quotent of H 1 E). Ths means that H E) s a subspace of H 1 E), thus to prove 2-smallness t wll suffce to show that H 1 s fntely generated n degrees 2. We begn by showng that H 1 s fntely generated n degrees 3. Let E be any set; the group H 1 E) has a bass {e j }, dual to the bass {x j } for H 1 E). Let j be elements of E, and consder the map E {1, 2, 3} takng to 1, j to 2, and everythng else to 3. The nduced map H 1 {1, 2, 3}) H 1 E) takes e 12 to e j, so we obtan a surjectve map from the projectve module P {1,2,3} to H 1 E). To get down from 3 to 2, consder the party map {1, 2, 3} {1, 2}. The nduced map H 1 {1, 2}) H 1 {1, 2, 3}) takes e 12 to e 12 + e 23. By symmetry, we can vary the map and obtan e 13 + e 23 and e 12 + e 13 as mages of nduced maps from H 1 {1, 2}) to H 1 {1, 2, 3}). Snce these three vectors span H 1 {1, 2, 3}), H 1 s generated n degree 2. For the last statement, we begn by notng that dm H 1 n) = n 2), therefore ) n r 2 H 1 ) = lm n 2 n = 0. 2 Ths mples r 2 H 1 ) = r 2H 1 ) = 0. Snce H H 1, we have r 2H ) = 0, as well. Remark 5.2. The second statement of Proposton 5.1 also follows from the fact that H s fntely generated as an FI-module [CEF15, Theorem 6.2.1]. More generally, they prove ths wth R 2 replaced by any connected, orented manfold of dmenson greater than 1 wth fnte dmensonal cohomology.) Ths mples that the dmenson of H n) grows as a polynomal n n [CEF15, Theorem 1.5], thus the same s true for the dmenson of the FS op -module H n) = H n). For any p 0, let Comp p, E) := = f:e [p+1] f:e [p+1] p+1 = ) H f 1 1)) H f 1 p + 1)) H 1 f 1 1)) H p+1 f 1 p + 1)). 11

12 It s clear that Comp p, comes endowed wth a natural FS op -module structure. Proposton 5.3. The FS op -module Comp p,0 s p + 1)-small, and Comp p, s p + 2)-small for all 1. Proof. By Lemma 4.2 and Proposton 5.1 the summand of Comp p, correspondng to the tuple 1,..., p+1 ) s d + 2)-small, where d s the number of k such that k = 0. When = 0, we have d = p + 1. When > 0, the maxmum value of d s p. 6 The man theorem For any fnte set E, let I E := {, j) j E}, and defne V E C I E the defnton of V n C In n a manner analogous to n Secton 6. In partcular, we have I [n] = I n and V [n] = V n. Defne the recprocal plane X E := XV E ), and let D E) := IH 2 X E ; C ). By Theorem 2.1, D E) s the th AutE)-equvarant Kazhdan-Lusztg coeffcent of the matrod M E assocated wth the complete graph on the vertex set E. In partcular, f we take E = [n], we have D n) = C Sn M n,. A surjectve map of sets E F s equvalent to the data of a partton of E along wth a bjecton between F and the set of parts of the partton. A partton of E determnes a flat of M E, and the bjecton between F and the set of parts of the partton determnes an somorphsm from X F to X V E ) F ). Thus, Theorem 3.31) gves us a map from D E) to D F ), and the frst half of Theorem 3.33) tells us that D s an FS-module. For any non-negatve ntegers p, q, defne A p,q E) := Comp p,2 p q E) D qp + 1). Snce Comp p,2 p q s an FS op -module wth an acton of the symmetrc group S p+1 gven by permutng the peces of the composton) and D q p + 1) s a fxed vector space equpped wth an acton of S p+1, A p,q nherts the structure of an FS op -module wth an acton of the symmetrc group := A p,q ) S p+1 be the nvarant submodule, and let B p,q ) be the dual FS-module. By Corollary 3.4, we have a frst quadrant cohomologcal spectral sequence wth E 1 page B p,q E) that converges to D E). By the second half of Theorem 3.33), each B p,q ) admts the structure S p+1. Let B p,q of an FS-module such that the FS-module maps commute wth the dfferentals n the spectral sequence. By Theorem 3.34), the FS-module structure on B p,q ) comng from Theorem 3.33) concdes wth the FS-module structure that we defned explctly. Theorem 6.1. For all 1, the FS op -module D s 2-small, and we have r 2 D ) = dm D 12). 2)! Proof. We frst prove that D s 2-small. Snce smallness s preserved under takng subquotents, t suffces to prove that B p,q s 2-small for all p and q. Snce B p,q A p,q, t suffces to prove t for 12

13 A p,q. By Proposton 5.3 and the fact that smallness s preserved by takng a tensor product wth a fxed vector space, A p,q s p + 1)-small when p + q = 2 and p + 22 p q))-small otherwse. Consder the case where p+q = 2. By defnton of the equvarant Kazhdan-Lusztg polynomal, D E) = 0 unless 2 < E 1 or E = 1 and = 0. In partcular, f p = 2 and q = 0, then D q p + 1) = D 2) = 0, and therefore A p,q = 0. Thus we may assume that p < 2. Snce A p,q s p + 1)-small t s also 2-small. Next, consder the case where p + q < 2, so A p,q s p + 22 p q))-small. By the above vanshng property for D E), we have D q p + 1) = 0 unless 2 q) < p or p = 0 and q =. Thus we may conclude that A p,q = 0 unless p + 22 p q) + p = 2 q) p + 2 < 2 or p = 0 and q =. In partcular, A p,q s 2-small, and therefore so s D. Ths argument n fact proves that A p,q s 2 1)-small unless p, q) = 0, ) or 2 1, 1), and the same s therefore true for B p,q. Furthermore, we have B 0, = H, and Proposton 5.1 tells us that r 2 H ) = 0. Thus r 2 B p,q ) = 0 unless p, q) = 2 1, 1), and Lemma 4.5 therefore tells us that r 2 D ) = r 2B 2 1,1 ). We have B 2 1,1 = Comp 2 1,0 ) S 2 D 1 2), where Comp 2 1,0) S 2 s the FS op -module that takes E to a vector space wth bass gven by parttons of E nto 2 nonempty peces. Ths means that dmcomp 2 1,0 ) S 2 n) s equal to the Strlng number of the second knd Sn, 2), thus r 2 D ) = r 2 B 2 1,1 ) = lm n and the theorem s proved. dm B 2 1,1 n) Sn, 2) dm D 1 2) 2) n = lm n 2) n = dm D 12), 2)! Let H u) := H D u) and G u) := G D u). Note that, snce representatons of fnte groups are self-dual, H u) and G u) may be regarded as generatng functons ordnary and exponental) for the degree Kazhdan-Lusztg coeffcents of brad matrods. The followng corollary follows mmedately from Theorems 4.1 and 6.1. Corollary 6.2. Let be a postve nteger. 1. If λ n and Hom Sn V λ, D n)) 0, then lλ) For any partton λ wth n λ + λ 1, dm Hom Sn Vλn), D n) ) s bounded by a polynomal n n of degree at most The ordnary generatng functon H u) s a ratonal functon whose poles are contaned n the set {1/j 1 j 2}. Furthermore, H u) has at worst a smple pole at 1/2. 4. There exsts polynomals p 0 u),..., p 2 u) such that the exponental generatng functon G u) s equal to d p j u)e ju. j=0 13

14 Furthermore, p 2 u) s equal to the constant polynomal wth value r 2 D ) = dm D 12) 2)!. Remark 6.3. Theorem 6.1 and Conjecture 2.4 combne to say that r 2 D ) = 2 3)!!2 1) 2 2)! = 2 1) 3 2.! In partcular, f Conjecture 2.4 s true or more generally f D 1 2) 0), then H u) does have a pole at 1/2. 7 Examples We now example the cases when = 1 or 2 n greater detal. Example 7.1. We frst consder the case when = 1. In [GPY17, Proposton 4.4], we showed that Hom Sn V λ, D 1 n)) = 0 for all λ wth more than 2 rows, and that dm Hom Sn V[k]n), D 1 n) ) s bounded by n/2 + 1 k. By [EPW16, Corollary 2.24], we have dm D 1 n) = 2 n 1 1 n 2), whch mples that u 4 H 1 u) = 1 u) 3 1 2u) and In partcular, r 2 D 1 ) = 1/2 = dm D 02)/2!. G 1 u) = 1 ) u e u e2u. Example 7.2. We next consder the case when = 2. By [EPW16, Corollary 2.24], we have dm D 2 n) = sn, n 2) Sn, n 1)Sn 1, 2) + Sn, 3) + Sn, 4), where sn, k) and Sn, k) are Strlng numbers of the frst and second knd, respectvely. We have well-known generatng functon denttes Sn, k)u n = n 1 u k k j=1 1 ju), as well as [Slo14, A000914] n 1 sn, n 2)u n = 2u3 + u 4 1 u) 5. Snce Sn, n 1)Sn 1, 2) = n 2) 2 n 2 1 ), t s not hard to show that Sn, n 1)Sn 1, 2)u n = n 1 u 2 1 2u) 3 u 2 1 u) 3. 14

15 Puttng t all together, we get H 2 u) = 2u3 + u 4 1 u) 5 u 2 1 2u) 3 u 2 ) 1 u) 3 u u)1 2u)1 3u) + u 4 1 u)1 2u)1 3u)1 4u) = 15u6 50u u 8 + 4u 9 1 u) 5 1 2u) 3 1 4u). After performng a partal fractons decomposton we fnd that r 4 D 2 ) = 1/24 = dm D 14)/4!. We do not have a general formula for the dmenson of Hom Sn V λ, D 2 n)), but we have computed D 2 n) for all n 9 [GPY17, Secton 4.4], and t s ndeed the case n these examples that the multplcty of V λ s D 2 n) s zero whenever λ has more than 4 rows. 8 The relatve case Let Γ be a fnte graph wth vertex set V. For any fnte set E, let ΓE) be the graph wth vertex set V E such that two elements of V are adjacent f and only f they were adjacent n Γ, and elements of E are adjacent to everythng. We wll defne an FS-module structure on the th AutE)-equvarant Kazhdan-Lusztg coeffcent D Γ E) of the matrod assocated wth the graph ΓE), and prove that the dual FS op -module s 2-small. If Γ s the empty graph, then ΓE) s just the complete graph on E, so we have D Γ = D. We begn by generalzng the materal n Secton 5. Let Γ = V, Q) be a fnte graph wth vertex set V and edge set Q, and let ConfΓ) be the set of maps from V to R 2 that send adjacent vertces to dstnct ponts. We have the followng descrpton of the cohomology rng of ConfΓ) [OT92, Theorems and 5.89]: / k H ConfΓ); C) = ΛC [x q ] q Q 1) j x q1 ˆx qj x qk j=1 q1,..., q k ) a closed path = the subrng of all meromorphc dfferental forms on C V generated by dz dz j z z j for all {, j} Q. By defnton, a map from Γ = V, Q) to Γ = V, Q ) s a map from V to V that takes Q to Q. Gven a map f : Γ Γ, we obtan a map H ConfΓ); C) H ConfΓ ); C) takng x q to x fq). In partcular, we obtan an FA-module HΓ E) := H ConfΓE)); C) and a dual FA op - module H ΓE) := H ConfΓE)); C). As n the case where Γ s empty, we can regard HΓ as an FS-module and H Γ as an FS op -module. The proof of the followng proposton s dentcal to the proof of Proposton 5.1. Proposton 8.1. The FS op -module H Γ 0 s 1-small. If 1, then HΓ s 2-small and r 2 H Γ ) = 0. 15

16 Gven a graph Γ wth vertex set V and a subset S V, let Γ S be the nduced subgraph wth vertex set S. Gven a surjectve map f : V V, let Γ f be the graph wth vertex set V whose edges are the mages of edges of Γ gnorng loops and multple edges). Fx a graph wth vertex set [p + 1], and defne Comp Γ, p, E) := f:v E [p+1] ΓE) f = connected j ΓE) f 1 j) H Conf ) ) ΓE) f 1)) 1 Conf ΓE)f 1 p+1) ; C. Gven surjectve maps g : E F and f : V F [p + 1] such that ΓE) f 1 j) s connected for all j, we can compose f wth g to obtan a surjectve map g f : V E [p + 1] wth the property that ΓE) g f) 1 ) s connected for all j and ΓE) g f = ΓF ) f. Ths observaton allows us to defne an FS op -module structure on Comp Γ, p,. Takng Γ to be the empty graph and the complete graph, we have Comp Γ, p, = Comp p,. The followng proposton generalzes Proposton 5.3. Proposton 8.2. The FS op -module Comp Γ, p,0 all 1. s p + 1)-small, and CompΓ, p, s p + 2)-small for Proof. Let Comp Γ p, := CompΓ, p,. We wll prove that CompΓ p, s p + 1)-small when = 0 and p + 2)-small when 1, and therefore so s each of ts summands. The above descrpton of the cohomology rng of ConfΓ) n terms of meromorphc dfferental forms makes t clear that H ConfΓ); C) s a subrng of H ConfV ); C), and therefore that the f-summand of Comp Γ, p, E) s a quotent of the f-summand of Comp p, V E). The proposton then follows from Proposton 5.3 and Lemma 4.3. We next generalze the materal n Secton 6. For any fnte set E and any non-negatve ntegers p, q, defne A p,q Γ, E) := Comp Γ, p,2 p q E) D q ). As n the case where Γ s the empty graph, A p,q Γ, s an FSop -module wth an acton of S p+1, and we defne the nvarant FS op -module B p,q Γ, := Ap,q ) S p+1 along wth ts dual FS-module B p,q Γ, ). There s agan a frst quadrant cohomologcal spectral sequence wth E 1 page B p,q Γ, E) that converges to D ΓE), nducng an FS-module structure on DΓ. Theorem 8.3. Let Γ be a graph wth vertex set V. 2-small, and we have For all 1, the FS op -module D Γ ) s r 2 D Γ ) ) = 2) V dm D 1 2) 2)! = 2) V r 2 D ). Proof. The same argument that we used n the proof of Theorem 6.1 shows that D Γ ) s 2-small 16

17 and r 2 D Γ ) ) = r 2 B 2 1,1 Γ, ). Explctly, we have B 2 1,1 Γ, E) = f:v E [2] D ΓE)f 1 ) S 2. When E s large, ΓE) f 1 j) s connected for all j and ΓE) f s equal to K 2 for almost all maps f : V E [2], and the number of such maps s asymptotc to 2) V +n. We therefore have and the theorem s proved. r 2 B 2 1,1 2) V +n dm D 1 2) Γ, ) = lm n 2) n 2)! = 2) V dm D 1 2), 2)! References [AB16] Federco Ardla and Adam Boocher, The closure of a lnear space n a product of lnes, J. Algebrac Combn ), no. 1, [Arn69] V. I. Arnol d, The cohomology rng of the group of dyed brads, Mat. Zametk ), [BGS96] [CEF15] Alexander Belnson, Vctor Gnzburg, and Wolfgang Soergel, Koszul dualty patterns n representaton theory, J. Amer. Math. Soc ), no. 2, Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stablty for representatons of symmetrc groups, Duke Math. J ), no. 9, [CEFN14] Thomas Church, Jordan S. Ellenberg, Benson Farb, and Roht Nagpal, FI-modules over Noetheran rngs, Geom. Topol ), no. 5, [EPW16] [Ged] [GPY17] [OT92] Ben Elas, Ncholas Proudfoot, and Max Wakefeld, The Kazhdan-Lusztg polynomal of a matrod, Adv. Math ), Kate Gedeon, Kazhdan-Lusztg polynomals of thagomzer matrods, arxv: Kate Gedeon, Ncholas Proudfoot, and Benjamn Young, The equvarant Kazhdan Lusztg polynomal of a matrod, J. Combn. Theory Ser. A ), Peter Orlk and Hroak Terao, Arrangements of hyperplanes, Grundlehren der Mathematschen Wssenschaften [Fundamental Prncples of Mathematcal Scences], vol. 300, Sprnger-Verlag, Berln, [Pr00] Temuraz Prashvl, Dold-Kan type theorem for Γ-groups, Math. Ann ), no. 2,

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