arXiv:2206.00324v1 [math.RT] 1 Jun 2022

KOSTKA NUMBERS AND FOURIER DUALITY

MICHAEL FINKELBERG, ALEXANDER POSTNIKOV, AND VADIM SCHECHTMAN

To Yuri Ivanovich Manin on his 85th birthday with admiration

May 26, 2022

Abstract. We relate the Fourier transform of perverse sheaves smooth alongthe coordinate hyperplane configuration in a complex vector space to theDeligne-Lusztig duality of unipotent representations of a general linear groupover a finite field. A similar relation is established for arbitrary finite Coxetergroups.

Contents

1. Introduction 21.1. 21.2. Acknowledgments 32. Hyperbolic sheaves over coordinate arrangements 32.1. Real coordinate arrangement 32.2. Complex coordinate arrangement 42.3. GGM sheaves 52.4. Fourier-Sato transform 52.5. Hyperbolic sheaves 62.6. Morita equivalence 62.7. Example n = 1 72.8. General n 82.9. Vanishing cycles at the origin: the Master complex 82.10. Explicit description of P 92.11. Amonodromic semisimple sheaves 93. Kostka numbers 103.1. Partitions and compositions 103.2. Young tableaux 113.3. Kostka numbers 113.4. Small Kostka numbers 123.5. Descents in Young tableaux 133.6. Skew Young diagrams 143.7. Young symmetrizers 14

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3.8. Induced modules and ribbon modules 153.9. Kostka numbers for Coxeter groups 173.10. Solomon’s decomposition of k[W ] 183.11. Descent basis 193.12. Proof of Theorem 3.11.2 204. Kostka sheaves 224.1. The sheaf F in GGM realization 224.2. The sheaf F in hyperbolic realization 224.3. A functor Rep(Sn+1)→ Hypass

n (Rep(Sn+1)) 234.4. Induced from parabolics as functions on flags 234.5. Induction and restriction 244.6. Parabolic complexes 244.7. Comparison with the vanishing cycles master complex 254.8. Relation to the permutohedron 254.9. Warning 265. From Sn+1 to GL(n+ 1,Fq) 265.1. Flag spaces and induced representations 265.2. Induction and restriction 275.3. Unipotent representations of GL(n+ 1,Fq) 28References 29

1. Introduction

1.1. As soon as perverse sheaves were discovered about 40 years ago, there ap-peared a problem of concrete linear algebraic description of abelian categoriesof perverse sheaves smooth along some particular stratifications (“How to glueperverse sheaves”). One important case is a complex affine space stratified by ahyperplane arrangement. In case such an arrangement is real, a solution of thisproblem was suggested in [KS] (hyperbolic sheaves).

In the present note we study a most trivial example of a hyperplane arrange-ment: namely the one of coordinate hyperplanes. In this case another descrip-tion of the category of perverse sheaves was proposed in [GGM] (iterated van-ishing/nearby cycles). It turns out that already in this simplest example, theinterplay between the two different descriptions of the same abelian categoryleads to some nontrivial combinatorial consequences.

Namely, we go even further along the route of simplification and considersemisimple perverse sheaves on Cn with trivial monodromy, however with co-efficients not in the category of vector spaces, but in the category C of repre-sentations of the symmetric group Sn+1 (over a field k of characteristic 0). Weconsider a perverse sheaf Fµ whose generic stalk is the irreducible Sn+1-module

KOSTKA NUMBERS AND FOURIER DUALITY 3

Vµ (for a partition µ of n + 1), and whose hyperbolic stalks are equal to induc-tion of Vµ to Sn+1 from appropriate parabolic subgroups. When µ = (n + 1),so that V(n+1) is trivial, we call the multiplicities of simple perverse sheaves inF(n+1) small Kostka numbers κλ,I . According to Theorem 3.5.1, κλ,I is equal tothe number of standard Young tableaux of shape λ with descent set I. We alsodiscuss an extension of (small) Kostka numbers to an arbitrary finite Coxetergroup W related to [So2].

The hyperbolic calculus developed in [FKS] allows to compute the Fourier-Satotransform of Fµ. Namely, we have FTFµ ∼= Fµt (transposed partition). Also, thehyperbolic calculus provides a master complex Mas•µ computing the vanishingcycles Φ0Fµ of Fµ at the origin. It turns out that Φ0Fµ ∼= Vµt , and the mastercomplex goes back at least to [So1].

For a prime power q, there is a Benson-Curtis isomorphism k[Sn+1]∼−→Hn+1

with the Iwahori-Hecke algebra. Thus C is equivalent to the category Cq of Hn+1-modules. Under this equivalence, the master complex Mas•µ corresponds to acomplex qMas•µ going back at least to [Kat]. According to loc.cit., a similarcomplex exists for the Hecke algebra of an arbitrary finite Coxeter group.

Furthermore, Cq is equivalent to the category of unipotent representations ofGL(n + 1,Fq), and under this equivalence, qMas•µ corresponds to the complexof [DL] defining the Alvis-Curtis-Deligne-Lusztig-Kawanaka duality [A, C, Kaw,DL].

1.2. Acknowledgments. This note was conceived in 2017. We are grateful toAnatol Kirillov and Mikhail Kapranov for the inspiring discussions. We alsothank George Lusztig for bringing [So2] to our attention.

M.F. was partially funded within the framework of the HSE University BasicResearch Program and the Russian Academic Excellence Project ‘5-100’.

2. Hyperbolic sheaves over coordinate arrangements

If (X,≤) is a poset, we call a bisheaf on X with values in a category C a coupleB = (γ, δ) of functors

γ : X → C, δ : Xop → C

such that for all x ∈ X, γ(x) = δ(x) =: B(x). Thus, for all x ≤ y we have twoarrows

γx,y : B(x) � B(y) : δy,x.

2.1. Real coordinate arrangement. Let V = Rn with coordinates x1, . . . , xn;

H = {H1, . . . , Hn},where Hi ⊂ V is given by the equation xi = 0.

S is the corresponding stratification of V .


The strata of S are in bijection with sequences

s = (s1, . . . , sn) ∈ {−1, 0, 1}n,

where Cs is given by n equalities and inequalities: xi = 0 if si = 0, and sixi > 0if si = ±1. So all the strata are cells, and in what follows we will refer to thestrata as cells.

To put it differently, the cells are numbered by couples (I, ε) where I is a subsetof [n] := {1, . . . , n}, and ε = (εj) ∈ {1,−1}[n]rI is a collection of signs on thecomplement. We denote the corresponding cell by C(I,ε).

We have C(I,ε) ≤ C(I′,ε′) (i.e. C(I,ε) lies in the closure of C(I′,ε′)) if and only if

(a) I ⊃ I ′, whence I := [n] r I ⊂ I ′,and(b) εj = ε′j for all j ∈ I.In other words, (I, ε) is obtained from (I, ε′) by replacing some elements ε′j in

the sequence ε′ by 0’s.Consider two cells Cs and C ′ = Cs′ where the sequence s′ is obtained from s by

replacing some element si = ±1 by s′i = −si. We call such two cells neighbours.Let C ′′ = Cs′′ where s′′ is obtained from s by replacing the element si by 0. ThisC ′′ is called the wall between C and C ′′.

Following Tits, let us call a gallery a sequence C0, . . . , Cr where all Ci, Ci+1 areneighbours. We call C0 and Cr faraway neighbours.

Each cell C of dimension p has 2p faraway neighbours (including itself), whoseunion is dense in L(C): the flat of C, the R-vector space spanned by C.

We denote by CI the positive cell C(I,ε) with all εj = 1. So, there are 2n positivecells.

2.2. Complex coordinate arrangement. Let

HC = {HC1 , . . . , H

Cn }

be the complex coordinate arrangement in VC = Cn, and let SC be the corre-sponding stratification of VC.

Its strata are SI , I ⊂ [n] := {1, . . . , n}, where

SI = {x ∈ VC|xi = 0 for i ∈ I, xi 6= 0 for i /∈ I}.

We write S ≤ S ′ iff S ⊂ S ′.In the complex case SI ≤ SJ iff J ⊂ I.A complex stratum SI with |I| = p is of real codimension 2p, and contains 2n−p

real cells C(I,ε).


2.3. GGM sheaves. We have two linear algebra descriptions of the categoryM(VC, SC) = Perv(VC, SC).

The first one is due to [GGM]. Namely, M(VC, SC) is equivalent to a categoryGGMn whose objects are certain bisheaves Φ on the poset SC with values in Vect,i.e.

— collections of 2n spaces

Φ(I) = Φ(SI) ∈ Vect, I ⊂ [n],

— and canonical maps

uIJ : Φ(I)→ Φ(J), I ⊂ J,

and variation maps

vJI : Φ(J)→ Φ(I), I ⊂ J,

satisfying the transitivity for triples I ⊂ J ⊂ K.They must satisfy the following further conditions.For all I, i ∈ I denote

vi = vI,Ir{i} : Φ(I) � Φ(I r {i}) : uIr{i},I = ui.

Then the mixed commutativity condition: uv = vu must hold, that is, for anyi, j ∈ I, i 6= j, the square

Φ(I r {i}) vj−−−→ Φ(I r {i, j})ui

y ui

yΦ(I)

vj−−−→ Φ(I r {j})must commute.

Also, the invertibility condition of 1 + vu must hold, that is for all i ∈ I ⊂ [n],the monodromy operator

Ti := 1 + viui : ΦI → ΦI

must be invertible.Note that 1 + viui is invertible if and only if 1 + uivi is invertible.This definition makes sense if we replace the coefficient category Vect by an

arbitrary additive category C; let us denote the corresponding GGM categoryGGMn(C).

2.4. Fourier-Sato transform. The Fourier-Sato transform Perv(VC, SC) ∼−→Perv(VC, SC) induces the same named auto-equivalence GGMn

∼−→GGMn. Moregenerally, for an arbitrary additive category C, we have an auto-equivalence

FT: GGMn(C) ∼−→GGMn(C)


that sends an object Φ = (Φ(I), u, v) to FT(Φ) = (Φ′(I) := Φ(I), u′, v′). Here weset for i 6∈ I ⊂ [n],

u′i := −vi : Φ(I)→ Φ(I r {i}), and v′i := ui(1 + viui)−1 : Φ(I r {i})→ Φ(I),

cf. [BK, Proposition 4.5] and [Mal, (II.2.1), (II.3.2)].

2.5. Hyperbolic sheaves. The second, “Mobius dual”, linear algebraic descrip-tion of M(VC, SC) is a version of [KS]. The category M(VC, SC) is equivalent to acategory Hypn of hyperbolic sheaves.

An object of Hypn is a bisheaf E with values in Vect on the poset of real strataS, i.e.

— a collection of 3n spaces E(C) = E(I, ε) ∈ Vect, C ∈ S,— maps γCC′ : E(C)→ E(C ′), δC′C : E(C)→ E(C ′), C ≤ C ′.These maps must satisfy the following conditions:Idempotence: For C ≤ C ′,

γCC′δC′C = IdE(C′).

Let C,C ′ be any pair of cells. There exists a cell C ′′ ≤ C and C ′′ ≤ C, forexample C ′′ = {0}. Define a map

φCC′ = γC′′CδCC′′ .

Due to the idempotence axiom this map does not depend on a choice of C ′′.Invertibility: If dimC = dimC ′ = p, and C,C ′ belong to the same linear

subspace L ⊂ V of dimension p then φCC′ is invertible.The invertibility axiom is equivalent to a seemingly weaker requirement:Weak invertibility: The map φCC′ = γC′′CδCC′′ is an isomorphism for all neigh-

bours.Mixed commutativity: γδ = δγ. That is, for C ≤ D, C ′ ≤ D′, C ≤ C ′, D ≤ D′,

the square

E(C ′)δ−−−→ E(D′)

γ

y γ

yE(C)

δ−−−→ E(D)

must commute.This definition makes sense if we replace the coefficient category Vect by an

arbitrary category C; let us denote the corresponding category Hypn(C).

2.6. Morita equivalence. We have equivalences of categories Lh : M(VC, SC) ∼−→Hypn (“hyperbolic, or big localization”) and Lg : M(VC, SC) ∼−→GGMn (“vanish-ing cycles, or small localization”), so there should exist equivalences of quiver


categories

P : Hypn∼−→GGMn, Q : GGMn

∼−→ Hypn.

Below we will construct equivalences

(2.6.1) P : Hypn(C) ∼−→GGMn(C), Q : GGMn(C) ∼−→ Hypn(C)

for an arbitrary additive category C.

2.7. Example n = 1. The construction below is a version of [KS, 9.A].An object of Hyp1(C) is a triple E = (E0, E±) of objects of C, and 4 maps

γ± : E0 → E±, δ± : E± → E0

such that γ±δ∓ are isomorphisms.An object of GGM1(C) is a couple Φ,Ψ ∈ (C) and 2 maps

v : Φ � Ψ : u

such that t := 1 + vu is an isomorphism.Let E ∈ Hyp1(C); we set Ψ := E+, Φ := Kerγ−, and v is the composition

Φ ↪→ E0γ+−→ Ψ,

while u is the composition

Ψδ+−→ E0

1+δ−γ−−−−−→ Φ.

This way we get an object

P (E) = (Φ,Ψ, v, u) ∈ GGM1(C).

In the opposite direction, given G = (Φ,Ψ, v, u) ∈ GGM1(C) with t := 1 + vu,we define

Q(G) = (E0, E±, γ±, δ±) ∈ Hyp1(C)

by

E+ = E− = Ψ; E0 = Φ⊕Ψ;

where δ− (resp. γ−) is given by the obvious inclusion (resp., projection), whereas

γ+(x, y) = −v(x) + y,

and

δ+(y) = (u(y), t(y)).


2.8. General n. We note that

GGMn(C) ∼= GGM1(GGMn−1(C)), and Hypn(C) ∼= Hyp1(Hypn−1(C)).

Whence we define Pn, Qn by induction: Pn being given as the composition

(2.8.1) Hypn(C) ∼= Hyp1(Hypn−1(C))P1−→ GGM1(Hypn−1(C))

Pn−1−−−→ GGM1(GGMn−1(C)) ∼= GGMn(C),

and similarly for Qn.

2.9. Vanishing cycles at the origin: the Master complex. We will give anexplicit description of the equivalence P : Hypn

∼−→ GGMn via vanishing cycles.Given E = (E(C), γ, δ) ∈ Hypn, we are going to describe the correspondingGGM-sheaf P (E) = (Φ(I), u, v).

First let us describe its vanishing cycles at the origin, Φ0 := Φ([n]). Geo-metrically, let M ∈ M(VC, SC) be the perverse sheaf corresponding to E; thesheaf of vanishing cycles Φf (M) corresponding to a function f(x) =

∑ni=1 xi is a

scyscraper concentrated at 0, and Φ0 is its stalk at 0:

Φ0 = Φf (M)0.

Algebraically the procedure of computing Φf (M) is described in [FKS], for anarbitrary real hyperplane arrangement. For the coordinate arrangement it reducesto the following.

We consider the positive cells CI , and denote E(I) := E(CI). Let S+ ⊂ S

denote the subset of positive cells.Let

Subin = {I ⊂ [n] : |I| = n− i} ∼= S+i := {C ∈ S+| dimC = i}.

We consider a complex

(2.9.1) Φ•(E) : 0→ E([n])→ ⊕I∈S+1 E(I)→ . . .→ E(∅)→ 0,

concentrated in degrees 0, . . . , n.The differentials are ±γ. More precisely, if J = I t j, then the JI matrix

coefficient of the differential is (−1)sγC(J)C(I), where s = |{i ∈ I : i < j}|.

Proposition 2.9.1. The complex Φ•(E) is acyclic in positive degrees, and

Φ0 = H0(Φ•(E)).

This is a particular case of [FKS, Theorem 2.3]. This complex Φ•(E) will becalled the Master complex of vanishing cycles for a hyperbolic sheaf E ∈ Hypn.


2.10. Explicit description of P . Similarly, for an arbitrary I ⊂ [n], |I| = p,let fI(x) =

∑i∈I xi. The sheaf of vanishing cycles

ΦfI (M)

is supported on the closure of the stratum S(I), and Φ(I) is the stalk of ΦfI (M)at CI . It is computed via an exact sequence

0→ Φ(I)→ E(I)→ ⊕I′⊂I,|I|=p−1E(I ′)→ . . .→ E(∅)→ 0,

where the differentials are ±γ.In particular, Φ(I) ⊂ E(I), Φ(I r {i}) ⊂ E(I r {i}), and the variation map

vi : Φ(I)→ Φ(I r {i}) (notation of §2.3) is induced by

γCICIr{i} : E(I)→ E(I r {i}).

Finally, the canonical map ui : Φ(I r {i})→ Φ(I) is induced by the composition

E(I r {i})δCIr{i}CI−−−−−−→ E(I)

1+δ−γ−−−−−→ E(I),

where E(I) = E(CI) = E(CI,ε)γ−−→ E(CIr{i},ε′)

δ−−→ E(CI,ε) = E(CI) = EI , andε′ consists of all positive signs except for one negative sign of i.

Remark 2.10.1. We will not give an explicit description of the equivalenceQ : GGMn

∼−→ Hypn; we just mention that for G = (Φ(I), u, v), the positivehyperbolic stalks of the corresponding E = Q(G) = (E(C), γ, δ) ∈ Hypn aregiven by

E(I) = ⊕J⊂IΦ(J)

(“Takeuchi formula”, cf. [KS, 4.C.(1)] and [T]).

2.11. Amonodromic semisimple sheaves. Fix a semisimple abelian categoryC. The category of perverse sheaves MC(VC, SC) = PervC(VC, SC) with coeffi-cients in C contains a full semisimple subcategory Pervass

C (VC, SC) formed by thedirect sums of constant sheaves along the strata closures. Under the equiva-lences PervC(VC, SC) ' Hypn(C) and PervC(VC, SC) ' GGMn(C) the subcategoryPervass

C (VC, SC) ⊂ PervC(VC, SC) corresponds to the full semisimple subcategoriesHypass

n (C) ⊂ Hypn(C) and GGMassn (C) ⊂ GGMn(C).

Namely, GGMassn (C) ⊂ GGMn(C) consists of all GGM sheaves (Φ(I), u, v) as

in §2.3 with all uIJ = vJI = 0.The description of Hypass

n (C) ⊂ Hypn(C) is a little bit more elaborate. Giventwo cells C(I,ε) ≤ C(I′,ε′) as in §2.1, we define the reflected cell C(I′′,ε′′) as follows.Recall from §2.1 that (I, ε) is obtained from (I ′, ε′) be replacing some elements ε′jin the sequence ε′ by 0’s. Now to obtain (I ′′, ε′′) we replace these elements by theopposite ones (i.e. change their signs). In particular, I ′′ = I ′, and dimC(I′,ε′) =dimC(I′′,ε′′).


Finally, a hyperbolic sheaf (E(C), γ, δ) lies in Hypassn (C) if for any cells C(I,ε) ≤

C(I′,ε′) and the corresponding reflected cell C(I′′,ε′′) we have

(2.11.1) KerγC(I,ε)C(I′,ε′)= KerγC(I,ε)C(I′′,ε′′)

and Im δC(I′,ε′)C(I,ε)= Im δC(I′′,ε′′)C(I,ε)

.

The equivalence P : Hypassn (C) ∼−→GGMass

n (C) takes (E(C), γ, δ) to (Φ(I), 0, 0),where Φ(I) :=

⋂J$I KerγCICJ .

The inverse equivalence Q : GGMassn (C) ∼−→ Hypass

n takes (Φ(I), 0, 0) to(E(C), γ, δ), where E(C(I,ε)) :=

⊕J⊂I Φ(J), while γ are given by the natural

embeddings of direct summands, and δ are given by the natural projections ontodirect summands.

Remark 2.11.1. The autoequivalence of §2.4

FT: GGMn(C) ∼−→GGMn(C)

induces an involutive autoequivalence

(2.11.2) FT: GGMassn (C) ∼−→GGMass

n (C), (Φ(I), 0, 0) 7→ (Φ(I), 0, 0).

The corresponding involutive autoequivalence

(2.11.3) FT: Hypassn (C) ∼−→ Hypass

n (C)

is a particular case of [FKS, Theorem 4.15].

3. Kostka numbers

In the next section we will study the categories of amonodromic semisimpleperverse sheaves Hypass

n (C), GGMassn (C) for the category C = Rep(Sn+1) of rep-

resentations of the symmetric group Sn+1 on n + 1 letters over a field k of char-acteristic 0.

In this section we prepare some basic material about Young tableaux, Kostkanumbers, and Rep(Sn+1). We also discuss a generalization of Kostka numbers toa finite Coxeter group W .

3.1. Partitions and compositions. A partition λ of m is a sequence of integers

λ = (λ1, . . . , λk), λ1 ≥ . . . ≥ λk > 0,∑

λi = m.

The Young diagram of a partition λ is the set {(i, j) ∈ Z2 | 1 ≤ i ≤ k, 1 ≤ j ≤ λi}.The elements (i, j) of a Young diagram, called boxes, are arranged on the planeand indexed as elements of a matrix, i.e., i is the row index of box (i, j) and jis its column index. Slightly abusing notation, we identify a partition λ with itsYoung diagram and denote the latter by the same letter λ. For example, we have

λ = (3, 3, 1) =


Let Pm be the set of partitions of m, equivalently, the set of Young diagramswith m boxes, and let P :=

⋃m≥0Pm.

A composition β of m is a sequence positive integers

β = (β1, . . . , βk) with∑i

βi = m.

Let Qm be the set of all compositions of m. We have an obvious inclusion

(3.1.1) ι : Pm ↪→ Qm

with a left inverse Qm → Pm, that associates to a composition its weakly de-creasing permutation.

3.2. Young tableaux. For a Young diagram λ ∈ P, a semi-standard Youngtableau T of shape λ is a filling of boxes of λ by positive integers such that theentries are weakly increasing along each row and strictly increasing along eachcolumn of T . The weight of T is a nonnegative integer sequence β = (β1, . . . , βk),where βi is the number of i’s in T . (Here we identify weights β obtained fromeach other by adding 0’s at the end: β = (β1, . . . , βk) = (β1, . . . , βk, 0, . . . , 0).)A standard Young tableau T a semi-standard Young tableau of weight β =(1, . . . , 1). We denote by SSYT(λ, β) the set of all semi-standard Young tableauxof shape λ and weight β. Also SYT(λ) denotes the set of all standard Youngtableaux of shape λ. For example, here is a semi-standard Young tableau T ∈SSYT((4, 3, 1), (2, 2, 3, 1)) and a standard Young tableau T ′ ∈ SYT((4, 3, 1)):

T = 1 1 2 32 3 34

T ′ = 1 2 4 73 5 68

Clearly, # SSYT(λ, β) = # SSYT(λ, β′), where β′ is the composition obtainedfrom weight β by removing all 0’s.

3.3. Kostka numbers. For a partition λ and a composition β, the Kostka num-ber Kλ,β is defined as the number of semi-standard Young tableaux of shape λand weight β:

Kλ,β := # SSYT(λ, β).

The Kostka numbers Kλ,β are known to be invariant under permutations ofparts of β. Thus, without loss of generality, the Kostka numbers can be labelled bya pair of partitions λ, µ ∈ P, Kλµ = Kλ,ι(µ), where ι denotes the inclusion (3.1.1).If µ = λ, the only semi-standard Young tableau of the same shape and weight λis obtained by filling the ith row of λ all all i’s, for i = 1, 2, . . . . So Kλλ = 1.


3.4. Small Kostka numbers. Let us fix a nonnegative integer n. Let [n] :={1, . . . , n}, and let Subn := 2[n] denote the set of subsets I ⊂ [n]. The set Subnis in bijection with the set Qn+1 of compositions of n+ 1.

Lemma 3.4.1. The following two maps % = %n : Subn∼→ Qn+1 and %−1 : Qn+1

∼→Subn are bijective and inverse to each other:

% : I = {i1 < i2 < . . . < ir} 7→ β = (i1, i2 − i1, . . . , ir − ir−1, n+ 1− ir),

%−1 : β = (β1, . . . , βk) 7→ I = {β1, β1 + β2, . . . , β1 + . . .+ βk−1}.

For example, %([n]) = (1, 1, . . . , 1) and %(∅) = (n+ 1).

Definition 3.4.2. For λ ∈ Pn+1 and I ∈ Subn, the small Kostka number κλ,I isdefined by

κλ,I :=∑J⊂I

(−1)|J |−|I|Kλ,%(J).

Proposition 3.4.3. For λ ∈ Pn+1 and I ∈ Subn, we have

(3.4.1) Kλ,%(I) =∑J⊂I

κλ,J ,

or, equivalently, Kλ,β =∑

J⊂%−1(β) κλ,J . This formula defines the numbers κλ,Iuniquely.

Proof. The claim follows from the Mobius inversion formula. �

Example 3.4.4. Let n = 2 and λ = (2, 1) ∈ P3. We have Kλ,ρ(∅) = Kλ,(3) = 0,Kλ,ρ({1}) = Kλ,(1,2) = 1, Kλ,ρ({2}) = Kλ,(2,1) = 1, Kλ,ρ({1,2}) = Kλ,(1,1,1) = 2. Thusthe small Kostka numbers are κλ,∅ = 0, κλ,{1} = 1 − 0 = 1, κλ,{2} = 1 − 0 = 1,κλ,{1,2} = 2− 1− 1 + 0 = 0.

Remark 3.4.5. For n ≤ 4, all small Kostka numbers κλ,I are either 0 or 1. Forn ≥ 5, the numbers κλ,I can be greater than 1. For example, for n = 3k − 1, wehave κ(2k−1,k,1),{k,2k} = k.

For λ ∈ Pn+1, let λt denote the conjugate partition of λ, i.e., the partitionwhose Young diagram is obtained from λ by transposition. For I ∈ Subn, letI := [n] r I.

Theorem 3.4.6. (1) The small Kostka numbers κλ,I are non-negative integers.

(2) The small Kostka numbers have the symmetry κλ,I = κλt,I .

This theorem follows from a combinatorial and a representation-theoreticalinterpretations of the small Kostka numbers κλ,I given below, see Theorem 3.5.1and Corollary 3.8.3.


3.5. Descents in Young tableaux. For a standard Young tableau T ∈ SYT(λ)of shape λ ∈ Pn+1, we say that i ∈ [n] is a descent of T if the entry i+1 is locatedin T below the entry i, i.e., the row containing i+ 1 is below the row containingi. The descent set Des(T ) := {i ∈ [n] | i is a descent of T} ∈ Subn, the set of alldescents of T .

Let T t denote the standard Young tableau obtained by transposing T . Clearly,the descent set of T t is the complement to the descent set of T : Des(T t) =[n] r Des(T ). For example, for

T = 1 2 4 83 5 76

and T t = 1 3 62 54 78

we have Des(T ) = {2, 4, 5} and Des(T ′) = [7] r {2, 4, 5} = {1, 3, 6, 7}.Theorem 3.4.6 follows immediately from the following claim.

Theorem 3.5.1. The small Kostka number κλ,I equals the number of standardYoung tableaux of shape λ with descent set Des(T ) = I.

Proof. Define the standardization map

std : SSYT(λ, β)→ SYT(λ)

as follows. For a semi-standard Young tableau T of weight β = (β1, . . . , βk), thestandardization std(T ) of T is the standard Young tableau obtained from T byreplacing its βi entries i by the integers in the interval [β1 + · · · + βi−1 + 1, β1 +· · ·+ βi] arranged in the increasing order from left to right, for i = 1, . . . , k. Forexample, we have

T = 1 1 2 32 3 34

7−→ std(T ) = 1 2 4 73 5 68

The standardization map gives a bijection between the set SSYT(λ, β) of allsemi-standard Young tableaux T of shape λ and weight β and the subset ofstandard Young tableaux T ′ ∈ SYT(λ) that have no descents in the union ofintervals

[1, β1 − 1] ∪ [β1 + 1, β1 + β2 − 1] ∪ · · · ∪ [β1 + · · ·+ βk−1 + 1, β1 + · · ·+ βk − 1],

or, equivalently, Des(T ′) ⊂ %−1(β) = {β1, β1 + β2, . . . , β1 + β2 + · · ·+ βk−1}:

std : SSYT(λ, β) ∼−→ {T ′ ∈ SYT(λ) | Des(T ′) ⊂ %−1(β)}.Let κλ,I be the number of standard Young tableaux of shape λ with descent

set I. We obtainKλ,β =

∑J⊂%−1(β)

κλ,J ,


which is exactly equation (3.4.1) that defines the small Kostka numbers. Thusκλ,I = κλ,I , as needed. �

3.6. Skew Young diagrams. For two Young diagrams λ, µ ∈ P such that λ ⊇ µ,i.e., λ contains µ as a subset, the skew Young diagram λ/µ is the set-theoreticdifference of the Young diagrams λ and µ. For example, for λ = (3, 3, 1) andµ = (2, 1), we have

λ/µ =

Usual Young diagrams can be considered as a special case of skew Young diagramsλ/µ with µ = ∅.

A connected component of a skew Young diagram λ/µ is a connected componentof the graph on the set of boxes (i, j) ∈ λ/µ with the edge set {((i, j), (i′, j′)) ||i − i′| + |j − j′| = 1}. For example, the above skew Young diagram has twoconnected components.

We consider two special types of skew Young diagrams: horizontal strips andribbons.1 A skew Young diagram λ/µ is a horizontal strip if each connectedcomponent of λ/µ consists of a single row of boxes. A skew Young diagram is aribbon if it contains no 2× 2 square and has exactly one connected component.

For a composition β = (β1, . . . , βk), let hstrip(β) be the horizontal strip whoseith row contains βi boxes, for i = 1, . . . , k (with rows touching each other at thecorners). Also let ribbon(β) be the ribbon whose ith row contains βi boxes, fori = 1, . . . , k. For example,

hstrip((2, 3, 1, 2)) = , ribbon((2, 3, 1, 2)) =

3.7. Young symmetrizers. Let k[Sn+1] be the group algebra of the symmetricgroup Sn+1. Let λ/µ be a skew Young diagram with n + 1 boxes. Let T beany filling of boxes of λ/µ by 1, . . . , n + 1 (without repetitions), i.e., T is any

bijection [n+ 1]∼→ {boxes of λ/µ}. (Here T is not necessarily a standard Young

tableau.) Let R(T ) (resp., C(T )) denote the subgroup of Sn+1 consisting of allpermutations preserving the rows (resp., the columns) of T .

The Young symmetrizer yT ∈ k[Sn+1] is defined as

aT =∑

w∈R(T )

w , bT =∑

w∈C(T )

ε(w)w , and yT = bT aT .

1Ribbons appear in the literature under many different names. They are MacMahon’s zigzagdiagrams, see [Mac]. They also called rim hooks, border strips, etc.


Here ε(w) ∈ {1,−1} denotes the sign of permutation w ∈ Sn+1.The left ideal

(3.7.1) LT := k[Sn+1] · yT ⊂ k[Sn+1]

is a representation of Sn+1.If T and T ′ have the same shape λ/µ, then yT = yT ′ · u, for some u ∈ Sn+1.

Thus LT ∼= LT ′ are isomorphic Sn+1-modules.The Specht module Vλ/µ is defined (up to an isomorphism) as

Vλ/µ := LT ,

for any filling T of the shape λ/µ. A choice of T gives an embedding of the Spechtmodule Vλ/µ as a left submodule LT of the group algebra k[Sn+1].

For µ = ∅, the Specht modules Vλ, λ ∈ Pn+1, are exactly (the isomorphismclasses of) all irreducible representations of the symmetric group Sn+1. We havethe following decomposition of the group algebra k[Sn+1] into a direct sum of leftsubmodules:

(3.7.2) k[Sn+1] =⊕

T∈SYTn+1

LT =⊕

λ∈Pn+1

⊕T∈SYT(λ)

LT

,

where SYTn+1 =⋃λ∈Pn+1

SYT(λ), see [We, Theorem 1.3.J].

3.8. Induced modules and ribbon modules. The semitic filling of a skewYoung diagram λ/µ is the filling T of boxes of λ/µ by 1, 2, 3, . . . reading the boxesby rows right-to-left top-to-bottom. For example, here are the semitic fillings ofthe horizontal strip hstrip((2, 3, 1, 2)) and the ribbon ribbon((2, 3, 1, 2)):

2 15 4 3

68 7

2 15 4 36

8 7

For a composition β = (β1, . . . , βk) of n + 1, define the induced module Mβ :=LT , where T is the semitic filling of the horizontal strip hstrip(β). Equivalently,

(3.8.1) Mβ := k[Sn+1] ·

∑w∈Sβ1×···×Sβk

w

∼= IndSn+1

Sβ1×···×Sβkk,

where the direct product of groups Sβ1 × · · · × Sβk is standardly embedded as asubgroup of Sn+1. For instance, M(1,...,1) = k[Sn+1] is the regular representation,and M(n+1) = k is the trivial representation of Sn+1.

It is well-known, cf. [Kos] and [FH, Corollary 4.39], that the Kostka numberKλ,β is exactly the multiplicity of the irreducible Sn+1-module Vλ in the induced


module Mβ:

(3.8.2) Kλ,β = dim HomSn+1(Mβ, Vλ) , equivalently, Mβ∼=

⊕λ∈Pn+1

V⊕Kλ,βλ .

Define the ribbon module Rβ as

(3.8.3) Rβ := LT ′ ,

where T ′ is the semitic filling of ribbon(β).Recall the bijection % : I 7→ β between subsets I ∈ Subn and compositions

β ∈ Qn+1, see Lemma 3.4.1. We can also label induced modules and ribbonmodules by subsets I ∈ Subn:

(3.8.4) MI := M%(I) and RI := R%(I).

Solomon [So2, Theorem 2] constructed a decomposition of the group algebraof a finite Coxeter group, see Theorem 3.10.1 below. For the symmetric groupSn+1, this decomposition can be described, as follows.

Theorem 3.8.1. [So2, Theorem 2] The group algebra k[Sn+1] decomposes intoa direct sum of ribbon modules:

(3.8.5) k[Sn+1] =⊕

I∈Subn

RI .

More generally, any induced module MI decomposes into a direct sum of ribbonmodules.

Theorem 3.8.2. For I ∈ Subn, the induced Sn+1-module MI decomposes into adirected sum of ribbon modules:

(3.8.6) MI =⊕J⊂I

RJ ,

This theorem is a special case of Corollary 3.11.3 below. Note that here wetreat all modules as concrete left submodules of the group algebra k[Sn+1] (andnot as equivalence classes of representations of Sn+1).

Corollary 3.8.3. For a partition λ ∈ Pn+1 and a subset I ∈ Subn, the smallKostka number κλ,I is the multiplicity of the irreducible Sn+1-module Vλ in theribbon module RI :

κλ,I = dim HomSn+1(RI , Vλ) , equivalently, RI∼=

⊕λ∈Pn+1

V⊕κλ,Iλ .

Proof. Let κλ,I be the multiplicity of Vλ in RI . Taking multiplicities of Vλ in allterms of the decomposition (3.8.6), we get Kλ,%(I) =

∑J⊂I κλ,J , which is exactly

equation (3.4.1) that defines the small Kostka numbers. Thus κλ,I = κλ,I , asneeded. �


Remark 3.8.4. For any skew Young diagram λ/µ, the multiplicities of irreduciblerepresentations Vν of the symmetric group in the Specht module Vλ/µ are calledthe Littlewood-Richardson coefficients cλµν :

Vλ/µ ∼=⊕ν

V⊗cλµνν .

The Littlewood-Richardson rule is a combinatorial rule for the cλµν . The combina-torial interpretation of the small Kostka numbers (Theorem 3.5.1) can be shownto be equivalent to the special case of the Littlewood-Richardson rule when λ/µis a ribbon.

Theorem 3.5.1, decomposition (3.7.2), and Corollary 3.8.3 imply the followingclaim. For I ∈ Subn, let SYT(λ, I) be the set of all standard-Young tableauxT of shape λ with n + 1 boxes with descent set Des(T ) = I. Also let SYTI :=⋃λ∈Pn+1

SYT(λ, I). According to Theorem 3.5.1, κλ,I = # SYT(λ, I).

Corollary 3.8.5. Let I ∈ Subn. The ribbon module RI is isomorphic to thedirect sum of irreducible submodules of k[Sn+1]:

RI∼=

⊕T∈SYTI

LT .

The induced module MI is isomorphic to the direct sum of irreducible submodulesof k[Sn+1]:

MI∼=⊕J⊂I

( ⊕T∈SYTJ

LT

).

3.9. Kostka numbers for Coxeter groups. The symmetric group Sn+1 is aCoxeter group. Some of the above constructions can be extended to an arbitraryfinite Coxeter group W .

Let W be a finite Coxeter group of rank n with Coxeter generators s1, . . . , sn.Let k[W ] be the group algebra of W over a field k of characteristic 0.

For a subset I ⊂ [n] := {1, . . . , n}, let WI denote the parabolic subgroup of Wgenerated by si, for i ∈ I. In particular, W[n] = W and W∅ = {1}.

For I ⊂ [n], define the induced module MI as the following left W -submoduleof the group algebra k[W ]

(3.9.1) MI := k[W ]

∑w∈I

w

∼= IndWWIk ,

where, as usual, I := [n] r I.If W = Sn+1, then the induced module MI is exactly the module Mβ

∼=Ind

Sn+1

Sβ1×···×Sβkk, where β = %(I), from §3.8.


By analogy with (3.8.2) and Definition 3.4.2, we define the W -Kostka numbersand the small W -Kostka numbers, as follows.

Definition 3.9.1. Let V be any irreducible representation of W , and let I ∈ Subnbe any subset of [n]. The W -Kostka number KV,I is the multiplicity of V in theinduced module MI :

(3.9.2) KV,I := dim Homk[W ](MI , V ).

The small W -Kostka number κV,I is the alternating sum of K-Kostka numbers:

(3.9.3) κV,I :=∑J⊂I

(−1)|J |−|I|KV,J .

Using Mobius inversion, we can express the K-Kostka numbers in terms of thesmall W -Kostka numbers as

(3.9.4) KV,I =∑J⊂I

κV,J .

Let kε be the sign representation ofW , which is the 1-dimensional k[W ]-module,where each Coxeter generator as si : x 7→ −x. For a W -module V , let

V t := V ⊗ kε .

The following claim generalizes Theorem 3.4.6.

Theorem 3.9.2. (1) The small W -Kostka numbers κV,I are non-negative inte-gers.

(2) The small Kostka numbers have the symmetry κV,I = κV t,I .

This theorem follows from Corollary 3.11.4 and Lemma 3.11.5 below. It wouldbe nice to find a simple combinatorial rule like Theorem 3.5.1 for calculation ofthe small W -Kostka numbers (e.g. when W is a Weyl group of type B or D).

3.10. Solomon’s decomposition of k[W ]. Define the following elements of thegroup algebra k[W ]

aI :=∑w∈WI

w and bI :=∑w∈WI

ε(w)w ,

where ε(w) = (−1)`(w) is the sign of w ∈ W and `(w) := min{l | w = si1 · · · sil}is the length of w.

Notice the induced module MI defined by (3.9.1) is MI = k[W ] aI .Define the ribbon W -module RI as

(3.10.1) RI := k[W ] bIaI .

If W = Sn+1, then the ribbon W -module RI is exactly the ribbon mod-ule (3.8.3) from §3.8.


Theorem 3.10.1. [So2, Theorem 2] The group algebra k[W ] decomposes into adirect sum of 2n ribbon modules:

k[W ] =⊕

I∈Subn

RI .

3.11. Descent basis. Let us give a k-linear basis of k[W ] which agrees withSolomon’s decomposition.

The (right) descent set Des(w) of an element w ∈ W is

Des(w) := {i ∈ I | `(w si) < `(w)}.For w ∈ W , define the following element of the group algebra

(3.11.1) dw := w bDes(w)a[n]rDes(w) ∈ k[W ].

Example 3.11.1. For W = S3 = {1, s1, s2, s1s2, s2s1, s1s2s1}, we have

d1 = 1 + s1 + s2 + s1s2 + s2s1 + s1s2s1,ds1 = s1(1− s1)(1 + s2) = −1 + s1 − s2 + s1s2,ds2 = s2(1− s2)(1 + s1) = −1− s1 + s2 + s2s1,ds1s2 = s1s2(1− s2)(1 + s1) = −1− s1 + s1s2 + s1s2s1,ds2s1 = s2s1(1− s1)(1 + s2) = −1− s2 + s2s1 + s1s2s1,ds1s2s1 = −1 + s1 + s2 − s1s2 − s2s1 + s1s2s1.

Theorem 3.11.2. (1) The set of elements {dw | w ∈ W} is a k-linear basis ofthe group algebra k[W ].

(2) For any I ∈ Subn, the set of elements {dw | w ∈ W such that Des(w) = I}is a k-linear basis of the ribbon module RI . In particular,

dimRI = #{w ∈ W | Des(w) = I}.

(3) For any I ∈ Subn, the set of elements {dw | w ∈ W such that Des(w) ⊂ I}is a k-linear basis of the induced module MI . In particular,

dimMI = #{w ∈ W | Des(w) ⊂ I}.A proof of this theorem is given in §3.12 below. It is immediate from the

definitions that dw ∈ RI if Des(w) = I, and that dw ∈MI if Des(w) ⊂ I.

Corollary 3.11.3. For I ∈ Subn, we have the decomposition of the inducedmodule MI into a direct sum of ribbon modules:

MI =⊕J⊂I

RJ .

Corollary 3.11.4. For an irreducible W -representation V and I ∈ Subn, thesmall W -Kostka number κV,I is the multiplicity of V in the ribbon module RI :

κV,I = dim Homk[W ](V,RI).


Proof. Let κV,I be the multiplicity of V in RI . By Corollary 3.11.3,

KV,I := dim Homk[W ](V,MI) =∑J⊂I

dim Homk[W ](V,RJ) =∑J⊂I

κV,I ,

which is exactly equation (3.9.4) defining the small Kostka numbers. Thus κV,I =κV,I . �

Recall that kε is the sign representation of W .

Lemma 3.11.5. We have the isomorphism of W -modules:

RI ⊗ kε ∼= RI .

Proof. By [So2, Lemma 12], we have the isomorphism k[W ] bJaI ∼= k[W ] aIbJ .Thus RI ⊗ kε = k[W ] aIbI

∼= k[W ] bIaI = RI . �

Now Theorem 3.9.2 follows from Corollary 3.11.4 and Lemma 3.11.5.

3.12. Proof of Theorem 3.11.2. For I ⊂ [n], define the (right) symmetrizationand the antisymmetrization operators SymI and AsymI acting on k[W ] by

SymI : f 7→ f aI and AsymI : f 7→ f bI

Let Im(SymI),Ker(SymI), Im(AsymI),Ker(AsymI) ⊂ k[W ] be the images andkernels of these operators.

Lemma 3.12.1. The ribbon module RI is isomorphic to the quotient module

RI∼= Im(AsymI)/(Ker(SymI) ∩ Im(AsymI)).

More precisely, the map SymI restricted to Im(AsymI) induces the isomorphismbetween Im(AsymI)/(Ker(SymI) ∩ Im(AsymI)) and RI .

Proof. We have RI := k[W ] bI aI = {f aI | f ∈ Im(AsymI)} = {SymI(f) | f ∈Im(AsymI)}. �

For i ∈ [n], define the subspaces Si := {f ∈ k[W ] | fsi = f} and Ai :={f ∈ k[W ] | fsi = −f} of the group algebra k[W ]. Clearly, Si = Im(Sym{i}) =

Ker(Asym{i}) and Ai = Im(Asym{i}) = Ker(Sym{i}).

Lemma 3.12.2. The image of the symmetrization operator SymI is the inter-section of subspaces:

Im(SymI) =⋂i∈I

Si,

and the kernel of SymI is the linear span of subspaces:

Ker(SymI) = Spani∈I

Ai.


Similarly, the image and the kernel of the antisymmetrization operator AsymI

are

Im(AsymI) =⋂i∈I

Ai and Ker(AsymI) = Spani∈I

Si.

Proof. An element f ∈ k[W ] belongs to the image of SymI if and only if f w = f ,for any w ∈ WI . Equivalently, we should have f si = f , for all i ∈ I, that is,f ∈

⋂i∈I S

i.An element g =

∑w∈W gw w ∈ k[W ] belongs to the kernel of SymI if and

only if the sum of its coefficients gw over any right WI-coset C in W is zero:∑w∈C gw = 0. Such g should be a linear combination of the elements w − wsi,

for w ∈ W and i ∈ I. This follows from the fact that the right WI-cosets in Ware exactly the connected components of the graph on set of vertices W with theset of edges {(w,wsi) | w ∈ W, i ∈ I}. For fixed i ∈ I, the elements w−w si, forw ∈ W , span the subspace Ai. Thus g should belong to the linear span of thesubspaces Ai over all i ∈ I.

The claim about the image and the kernel of AsymI is proved analogously. �

For w ∈ W , let

cw := w bDes(w) ∈ k[W ].

Lemma 3.12.3. The set {cw | w ∈ W} is a k-linear basis of k[W ].

Proof. Notice that w is the unique maximal by length element in the expansionof cw. Thus the set of elements {cw|w ∈ W} is related to the standard linearbasis {w | w ∈ W} of k[W ] by a triangular matrix with 1’s on the diagonal. Thus{cw | w ∈ W} is a linear basis of k[W ]. �

Lemma 3.12.4. In the linear basis {cw | w ∈ W}, the subspace Ai is the co-ordinate subspace of k[W ] spanned by the basis elements cw, for all w such thati ∈ Des(w):

Ai = Spanw | i∈Des(w)

cw.

Moreover, for any I ⊂ [n], Im(AsymI) and Ker(SymI) are the coordinate sub-spaces given by

Im(AsymI) = Spanw | I⊂Des(w)

cw and Ker(SymI) = Spanw | I∩Des(w) 6=∅

cw.

Proof. Clearly, cw ∈ Ai if i ∈ Des(w). Since #{w ∈ W | i ∈ Des(w)} =|W |/2 = dimAi, we deduce that the basis elements cw such that i ∈ Des(w) spanthe subspace Ai. Now the claims about Im(AsymI) and Ker(SymI) follow fromLemma 3.12.2. �


Now we can prove part (2) of Theorem 3.11.2. By Lemmas 3.12.1 and 3.12.4,the map SymI , restricted to Span {cw | I ⊂ Des(w)}, induces the isomorphism(

Spanw | I⊂Des(w)

cw

)/

(Span

w | I⊂Des(w), I∩Des(w) 6=∅cw

)∼−→RI .

Thus the elements SymI(cw), for all w ∈ W such that Des(w) = I, form a linearbasis of RI . This is exactly the claim of Theorem 3.11.2(2), because SymI(cw) =w bDes(w)aI =: dw.

Part (1) of Theorem 3.11.2 follows from part (2) and Solomon’s decomposition(Theorem 3.10.1).

To prove part (3) Theorem 3.11.2, notice that dw ∈ MI , for all w ∈ W suchthat Des(w) ⊂ I. Such w’s are exactly the minimal length coset representativesfor the right cosets W/WI . So we get |W |/|WI | linearly independent elementsdw in the space MI

∼= IndWWIk of dimension |W |/|WI |. Thus they form a k-linear

basis of MI .

4. Kostka sheaves

Let C = Rep(Sn+1) = k[Sn+1]-mod denote the category of finite dimensionalrepresentations of Sn+1 over k, that is the category of finite dimensional leftk[Sn+1]-modules. The key object of this section is a semisimple amonodromicperverse sheaf F smooth along the coordinate stratification of Cn. Namely, it isa direct sum of constant sheaves FI on the strata closures SI , I ⊂ Subn. Finally,

FI =⊕

λ∈Pn+1

V⊕κλ,Iλ [n− |I|].

4.1. The sheaf F in GGM realization. The corresponding amonodromicsemisimple GGM sheaf with values in C is

P = P(n) ∈ GGMassn (Rep(Sn+1)).

By definition, for any I ∈ Subn we have P(I) = RI , the ribbon module definedby (3.8.3) and (3.8.4).

4.2. The sheaf F in hyperbolic realization. The corresponding amonodromicsemisimple hyperbolic sheaf with values in C is

Q = Q(n) ∈ Hypassn (Rep(Sn+1)).

Explicitly, by §2.11 its hyperbolic stalks are

Q(C(I,ε)) = ⊕J⊂IRJ .

Therefore, by Theorem 3.8.2, they are isomorphic to the induced modules

(4.2.1) Q(C(I,ε)) = Q(I) 'MI = M%(I)


We recall that % denotes the isomorphism % : Subn∼−→Qn+1 of Lemma 3.4.1, so

that %(I) = α is a composition of n+1, and Mα is the Sn+1-module induced fromthe trivial representation of the subgroup Sα ⊂ Sn+1.

4.3. A functor Rep(Sn+1) → Hypassn (Rep(Sn+1)). More generally, for any M ∈

Rep(Sn+1) we define a hyperbolic sheaf

QM = (M ⊗Q(C(I,ε)), IdM ⊗ γ, IdM ⊗ δ).This way we get a “localization” functor

Q : Rep(Sn+1)→ Hypassn (Rep(Sn+1)).

Theorem 4.3.1. For M ∈ Rep(Sn+1), we have an isomorphism FT QM 'QM⊗kε.

Proof. It suffices to construct an isomorphism FT Q ' Q ⊗ kε. Equivalently, inthe GGM realization, we have to construct an isomorphism FT P 'P⊗kε. Theexistence of the desired isomorphism follows immediately from the isomorphismVλ⊗kε ' Vλt , Theorem 3.4.6, and the formula for Fourier-Sato transform in §2.4.

�

4.4. Induced from parabolics as functions on flags. Let us call a type asequence of integers χ = (χ1, . . . , χp) with

1 ≤ χ1 < . . . < χp ≤ n+ 1.

Notation: {χ} = {χ1, . . . , χp}.The number p = `(χ) will be called the length of χ. The set of types of length

p will be denoted Typp; this set contains(n+1p

)elements.

To each type corresponds a composition of n+ 1

(4.4.1) α(χ) = (χ1, χ2 − χ1, . . . , n+ 1− χp) ∈ Qn+1

A flag of type χ is a chain I• of subsets I1 ⊂ . . . ⊂ Ip ⊂ [n + 1] with |Ii| = χi,and p is the length of I•.

We denote by F`χ the set of all flags of type χ, and by F`p the set of all flagsof length p.

The set F`χ is acted upon from the left by Sn+1 and is isomorphic to F`χ ∼=Sn+1/Sχ, where a parabolic subgroup Sχ is the stabilizer of the standard flag

[χ1] ⊂ . . . ⊂ [χp].

We haveSχ ∼= Sα1 × Sα2 × . . .× Sαp+1 = Sα,

where α = α(χ), cf. (4.4.1).Let Mχ denote the space of maps of sets

MapsSets(F`χ, k) = {f : F`χ → k};


it is an Sn+1-representation isomorphic to Mα introduced §3.8.This realization of the modules Mα is convenient for describing some natural

morphisms between them.For instance, there are (n + 1)! flags of length n + 1, all of them having type

χ0 = (1, 2, . . . , n + 1). The corresponding representation Mχ0 is the regular one.We postulate that there is a single type ∅ of length zero, and define M∅ to be thetrivial representation k.

4.5. Induction and restriction. For two types χ, θ we write θ ⊂ χ if {θ} ⊂ {χ}.We have obvious maps ∂χθ : F`χ → F`θ, wherefrom we obtain the “restriction”,

or pullback, morphisms in Rep(Sn+1)

rθχ = ∂∗χθ : Mθ →Mχ.

Their conjugate are “induction”, or pushout, morphisms

iχθ = ∂χθ∗ : Mχ →Mθ

are explicitly defined as follows: for f ∈Mχ, I• ∈ F`θ,

(4.5.1) iχθ(f)(I•) =∑

J•∈∂−1χθ (I•)

f(J•)

4.6. Parabolic complexes. These are two dual complexes of length n inRep(Sn+1).

The master complex going back at least to [Kat, (1.2)] (specialized to the casewhen the Weyl group W = Sn+1, the module M is trivial, and q = 1) is

(4.6.1) Mas• : 0→Mχ0 → ⊕χ:|χ|=n−1Mχ → . . .→ ⊕χ:|χ|=1Mχ →M∅ → 0

The differentials

dp : ⊕χ:|χ|=pMχ → ⊕χ:|χ|=p−1Mχ

are induced by the maps iχθ with appropriate signs. More precisely, their nonzeromatrix elements are the maps dχχ′ : Mχ →Mχ′ with χ′ ⊂ χ, |χ′| = |χ|−1. Givena type χ of length p, there are p subtypes χ′ = ∂iχ, 1 ≤ i ≤ p, of length p− 1.

By definition, dχ,∂iχ = (−1)iiχ,∂iχ. We consider the “master complex” (4.6.1)as concentrated in degrees [0, n].

The conjugate master complex is a similar one with differentials induced by therestriction maps rχ′,χ, |χ′| = |χ| − 1 (with signs):

(4.6.2) Mas′• : 0→M∅ → ⊕χ:|χ|=1Mχ → . . .→Mχ0 → 0

By [Kat, (#) at page 942] applied to the case when the Weyl group W = Sn+1,the module M is trivial, and q = 1,

(4.6.3) H i(Mas•) = 0, i > 0,


and the only nonzero cohomology is

(4.6.4) H0(Mas•) = kε,

the sign representation.

Corollary 4.6.1. H i(Mas′•) = 0, i > 0.

4.7. Comparison with the vanishing cycles master complex. Recall thecomplex Φ•(Q) introduced in (2.9.1) for arbitrary hyperbolic sheaf (E, γ, δ).

Theorem 4.7.1. There exists an isomorphism of complexes in Rep(Sn+1)

(4.7.1) Mas• ∼−→ Φ•(Q).

Proof. We know already that the individual terms of the above complexes areisomorphic, see (4.2.1). Both of them start with the regular representation M[n] =M(11...1) = k[Sn+1]. By (4.6.4), H0(Mas•) = kε.

On the other hand, H0(Φ•(Q)) ∼= kε. Indeed, the Euler characteristic of Φ•(Q)is kε e.g. by [So1, Theorem 2], and H>0(Φ•(Q)) = 0 by Proposition 2.9.1.

We will apply the following elementary

Lemma 4.7.2. Let C be a semisimple category, A ⊂ B,A′ ⊂ B′ objects of C. IfA ∼= A′ and B ∼= B′ then A/B ∼= A′/B′. �

We can reformulate the lemma as follows. Choose an isomorphism φA : A ∼−→A′.Since C is semisimple, there are objects C,C ′ such that B ∼= A ⊕ C and B′ ∼=A′ ⊕ C ′. According to the lemma, C ∼= C ′. Whence there exists an isomorphismφB : B ∼−→B′ making the square

A −−−→ B

φA

y φB

yA′ −−−→ B′

commutative.Now we can construct the desired isomorphism Mas• ∼−→ Φ•(Q) inductively

from left to right, using the isomorphism H0(Φ•(Q(n))) ∼= kε and the acyclicity ofboth complexes in positive degrees. �

4.8. Relation to the permutohedron. We present a geometric interpretationof the master complex of vanishing cycles Φ•(Q), cf. [Kat, So1, Wi].

Proposition 4.8.1. The complex Φ•(Q) is isomorphic, as a complex of k-vectorspaces, to the complex C•(Permn) of cochains of the n-th permutohedron.

For instance, let n = 2. The complex Φ•(Q) = Φ•(Q(2)) has the form

0→ k[S3]→ k[S ′]⊕ k[S ′′]→ k→ 0.


Here S ′, S ′′ ∼= S2 are the subgroups of S3 generated by transpositions s1 = (12)and s2 = (23) respectively. So the dimensions of its terms are 6, 6, 1, and it isisomorphic to the complex of cochains of the hexagon Perm2.

Proof. Our complex Φ•(Q) has the form

0→ k[W ]→ ⊕i∈Sk[W/W{i}]→ ⊕{i,j}⊂Sk[W/W{i,j}]→ . . .→ k→ 0,

where W = Sn+1, and for a subset I ⊂ S of the set of Coxeter generators,WI ⊂ W denotes the subgroup generated by si, i ∈ I.

On the other hand, consider the root arrangement in RS; the permutohedronPW is a polygon dual to this arrangement. Thus, its vertices are in one-to-one correspondence with the chambers that form a set on which W acts simplytransitively, i.e. after choosing a fundamental chamber C we can identify this setwith W , i.e. with a base of k[W ].

Next, the edges of PW are in bijection with walls; there are n walls Mi onthe boundary of C which correspond to simple reflections si; the stabilizer of Mi

being the subgroup W{i}. So the set of walls is in bijection with∐

i∈SW/W{i},etc. �

4.9. Warning. One might think that the maps iχθ and rθχ give rise to a hy-perbolic sheaf, but this is not so: the collinearity axiom does not hold, as theexample of S3 already shows.

Namely, let n = 2. Consider the square

M(1)r−−−→ M∅

i

y i

yM(12)

r−−−→ M(2).

Let f ∈ M(1), i.e. f is a function from one-element subsets of [3] to k. Then

r(f) =∑3

i=1 f(i), and ir(f) is the constant function with value r(f).On the other hand, i(f)({i} ⊂ {i, j}) = f(i), and ir(f)({i, j}) = f(i) + f(j).

Thus, ir 6= ri, the square is not commutative.

5. From Sn+1 to GL(n+ 1,Fq)

We denote G = Gn+1 = GLn+1(Fq).

5.1. Flag spaces and induced representations. We fix a vector space U =Fn+1q . For a type χ = (χ1, . . . , χp) ∈ Typp (notation of §4.4), a flag of type χ is a

chain of linear subspaces

F• : 0 = U0 ⊂ U1 ⊂ . . . ⊂ Up ⊂ U ;

such that dimUi = χi. We will call p the length of F•.


We denote by F`χ(q) the set of all flags of type χ, and by F`p(q) the set of allflags of length p.

The set F`χ(q) is acted upon from the left by G and is a homogeneous setF`χ(q) ∼= G/Pχ, where Pχ is the standard parabolic subgroup, the stabilizer ofthe standard flag Fχ1

q ⊂ . . . ⊂ Fχpq .For χ = (1, 2, . . . , n, n+ 1), we have Pχ = B, the standard Borel.Let Mχ(q) denote the space of maps of sets

MapsSets(F`χ(q), k) = {f : F`χ(q)→ k};it is a G-module isomorphic to IndGPχk.

5.2. Induction and restriction. We have the natural q-analogues of the mastercomplex and its conjugate of §4.6.

Let θ ⊂ χ be two types. We have obvious maps

∂χθ : F`χ(q)→ F`θ(q),

wherefrom the restriction, or pullback, morphisms in Rep(G)

rθχ = ∂∗χθ : Mθ(q)→Mχ(q).

Their conjugate are induction, or pushout, morphisms

iχθ = ∂χθ∗ : Mχ(q)→Mθ(q),

that are explicitly defined as follows: for f ∈Mχ(q), F• ∈ F`θ,

(5.2.1) iχθ(f)(F•) =∑

F ′•∈∂−1χθ (F•)

f(F ′•).

Proceeding as in §4.6 we obtain a parabolic complex

(5.2.2) DL(k)• : 0→Mχ0(q)→ ⊕χ:|χ|=n−1Mχ(q)→ . . .→M∅(q)→ 0

This is nothing but the Deligne-Lusztig complex [DL, (1.2)] for the case of trivialrepresentation E = k of GL(n+ 1,Fq). According to [DL, Theorem in §2],

(5.2.3) H i(DL(k)•) = 0, i > 0.

The only nonzero cohomology is H0(DL(k)•) = St, the Steinberg module [St].

5.2.1. Example n = 1. Consider the following hyperbolic sheaf E ∈M(C, 0) withvalues in Rep(GL(2,Fq)):

E0 = IndGBk = Maps(P1, k), E+ = E− = k.

We haveIndGBk

∼= k⊕ St .

The Fourier-Sato transform FT(E) has the hyperbolic stalks

E∨0 = E0 = IndGBk, E∨+ = E∨− = St .


5.2.2. Example n = 2. We have 4 = 22 standard parabolics G,P1, P2, B with

G/G = {∗}, G/P1 = P2, G/P2 = P2∨.

The real stratification SR of R2 has 4 2-dimensional cells C±i , i = 1, 2; and 41-dimensional cells `±i , i = 1, 2, and the 0-dimensional cell {0}.

We define a hyperbolic sheaf E ∈M(C2, S) with hyperbolic stalks

EC±i = k,

E`±1 = IndGP1k = Map(P2, k), E`±2 = IndGP2

k = Map(P2∨, k),

E0 = IndGBk = Map(G/B, k).

We have the decomposition into irreducibles

E0∼= k⊕ St′⊗k2 ⊕ St .

We have dim St′ = q2 + q, dim St = q3, and |G/B| = q3 + 2q2 + 2q + 1.The Fourier-Sato transform FT(E) has hyperbolic stalks

E∨0 = EC±i , E∨`±1

= E`±2 , E∨`±2

= E`±1 , E∨C±i

= E0.

5.3. Unipotent representations of GL(n + 1,Fq). Let Hn+1 denote theIwahori-Hecke algebra of Sn+1 at q. There is an isomorphism [BC] of algebrask[Sn+1] ' Hn+1. At the price of extending k by

√q, one can choose an explicit

isomorphism of [Lus]. It gives rise to an equivalence of semisimple abeliancategories C = Rep(Sn+1) ∼= Rep(Hn+1) =: Cq. Under this equivalence, themaster complex Mas• goes to the complex qMas• of [Kat, (1.2)] (specialized tothe case when the Weyl group W = Sn+1, and the representation M is trivial).

Furthermore, Cq is canonically equivalent to the semisimple abelian categoryCG := Repunip(Gn+1) of unipotent representations of GLn+1(Fq): direct sums ofconstituents of the natural representation of GLn+1(Fq) in the space of functionson the flag variety. Under this equivalence, qMas• goes to the Deligne-Lusztigcomplex [DL] DL(k) of the trivial G-module k.

More generally, composing the functor Q of §4.3 with the above equivalenceCG ' C, we obtain a functor

QG : Repunip(Gn+1)→ Hypassn (Repunip(Gn+1)).

Now given a unipotent representation K ∈ Repunip(Gn+1), we consider the com-

plex of vanishing cycles Φ•(QG(K)) of (2.9.1). Then it follows from Theorem 4.7.1that Φ•(QG(K)) is isomorphic to the Deligne-Lusztig complex DL(K). Finally, itfollows from Theorem 4.3.1 that we have an isomorphism FT QG(K) ' QG(K∨),where K∨ stands for the Deligne-Lusztig dual of K.


References

[A] D. Alvis, The duality operation in the character ring of a finite Chevalley group,Bull. Amer. Math. Soc. 1 (1979), 907–911.

[BC] C. T. Benson, C. W. Curtis, On the degrees and rationality of certain charactersof finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273.

[BK] R. Bezrukavnikov, M. Kapranov, Microlocal sheaves and quiver varieties, Ann.Fac. Sci. Toulouse Math. (6) 25 (2016), no. 2–3, 473–516.

[C] C. W. Curtis, Truncation and duality in the character ring of a finite group of Lietype, J. Algebra 62 (1980), 320–332.

[DL] P. Deligne, G. Lusztig, Duality for representations of a reductive group over a finitefield, J. Algebra 74 (1982), 284–291.

[FH] W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics 129(1991).

[FKS] M. Finkelberg, M. Kapranov, V. Schechtman, Fourier transform on hyperplanearrangements, arXiv:1712.0742.

[GGM] A. Galligo, M. Granger, Ph. Maisonobe, D-modules et faisceaux pervers dont lesupport singulier est un croisement normal, Ann. Inst. Fourier 35 (1985), 1–48.

[Kat] S.-I. Kato, Duality for representations of a Hecke algebra, Proc. Amer. Math. Soc.19 (1993), 941–946.

[Kaw] N. Kawanaka, Fourier transforms of nilpotently supported invariant functions ona simple Lie algebra over a finite field, Invent. Math. 69 (1982), no. 3, 411–435.

[Kos] C. Kostka, Uber den Zusammenhang zwischen einigen Formen von symmetrischenFunktionen, J. reine und angew. Math. 93 (1882), 89–123.

[KS] M. Kapranov, V. Schechtman, Perverse sheaves over real hyperplane arrangements,Ann. Math. 183 (2016), 619–679.

[Lus] G. Lusztig, On a Theorem of Benson and Curtis, J. Algebra 71 (1981), 490–498.[Mac] P. A. MacMahon, Combinatory Analysis, Cambridge 1915–1916.[Mal] B. Malgrange, Equations Differentielles a Coefficients Polynomiaux, Progress in

Math. 96, Birkhauser, Boston (1991).[So1] L. Solomon, The orders of finite Chevalley groups, J. Algebra 3 (1966), 376–393.[So2] L. Solomon, A Decomposition of the Group Algebra of a Finite Coxeter Group, J.

Algebra 9 (1968), 220–239.[St] R. Steinberg, A geometric approach to the theory of representations of the full linear

group over a Galois field, Trans. Amer. Math. Soc. 71 (1951), no. 2, 274–282.[T] K. Takeuchi, Dimension formula for hyperfunction solutions to holonomic D-

modules, Adv. Math. 180 (2003), 134–145.[We] H. Weyl, The classical groups, their invariants and representations, Princeton

Landmarks in Mathematics, Princeton University Press, Princeton, NJ (1997),15th printing, Princeton Paperbacks.

[Wi] E. Witt, Spiegelungsgruppen und Aufzahlung halbeinfacher Liecher Ringe, Abh.Math. Sem. Univ. Hamburg 14 (1941), 289–322.

http://arxiv.org/abs/1712.0742


National Research University Higher School of Economics, Russian Federa-tion, Department of Mathematics, 6 Usacheva St, 119048 Moscow;Skolkovo Institute of Science and Technology;Institute for the Information Transmission Problems

Email address: [email protected]

Department of Mathematics, Massachusetts Institute of Technology, 77 Mas-sachusetts Ave, Cambridge MA 02139, USA


Institut de Mathematiques de Toulouse, Universite Paul Sabatier, 118 Routede Narbonne, 31062 Toulouse, France


arXiv:2206.00324v1 [math.RT] 1 Jun 2022

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