Connecting decomposition numbers

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 June 2014

Abstract.

Introduction

This paper tries to figures out what the heck is going on in all these different Hecke algebra modular representation worlds.

Kazhdan-Lusztig polynomials and canonical bases

2.1.1 Let W be a finite Weyl group and let (w) be the length of wW. The Iwahori-Hecke algebra H is the algebra with basis Tw, wW, and multiplication determined by the relations TwTw= Tww,if (ww)=(w) +(w),and Tsi2=v-2 +(v-2-1) Tsi, (2.1) for simple reflections si in W. Setting Hw=v(w)Tw, for wW, we have Hsi2=1+(v-1-v)Hsi. Define a -linear involution A:HH by v=v-1 andHx= (Hx-1)-1. (2.2) The Kazhdan-Lusztig polynomials Py,x(v) for W are defined by finding the unique elements H_x, xW, of H such that H_x= H_x,and H_x=yW Py,x(v) Hywith Py,xv[v]. These are renormalized versions of the original Kazhdan-Lusztig polynomials Py,x(q) defined in [KL1]: If q=v-2 then Py,x(v)=v(x)-(y)Py,x(q). Note that Pw,w(v)=0 unlessww.

2.1.2 If W is replaced by an affine Weyl group W then ??? defines the affine Hecke algebra H and the resulting polynomials Py,x(v), y,xW, are the Kazhdan-Lusztig polynomials for the affine Weyl group W.

2.1.3 Let W be an affine Weyl group and let W be the corresponding finite Weyl group. Let 𝒜+ be the set of alcoves for the hyperplane arrangement of W which are in the dominant chamber. The minimal length representatives of cosets in W\W are in bijection with the alcoves in the dominant chamber via xxA+, where A+ is the fundamental alcove for the affine Weyl group. The "sign" representation of H is given by Hs-v (i.e. Ts-1) and the H module IndHH(sgn) =HHsgn, has basis NA=NxA+=Nx=Hx1, indexed by the alcoves A in 𝒜+. Define an involution on 𝒩 by ??? and N1=N1and hn=h n,forn𝒩 andhH. For each A𝒜+ there is a unique element N_A𝒩 such that N_A= N_A,and N_A=B𝒜 nB,A(v)NB withnB,A(v) v[v]. The polynomials nB,A(v) are the parabolic Kazhdan-Lusztig polynomials for H corresponding to the sign representation of H. Soergel [Soe1998, Prop 3.4] proves the following identity of Deodhar [Dd], ny,x(v)= zW(-v)(z) Pzy,x(v). (2.3)

The canonical basis

2.2.1 Define εi=Λi- Λi-1, 1i,α0 =ε-ε1+δ, andαi=εi -εi+1, 1i-1.

2.2.2 The negative part of the quantum group associated to the graph is the algebra defined by generators Fi, indexed by the nodes of the graph, and by relations p=0-aij+1 [-aij+1]! [p]![-aij+1-p]! FjpFiFj-aij+1-p =0,forij, where -aij is the number of edges connecting i and j in the graph.

2.2.3 These positive roots for U(𝔰𝔩ˆ) are εi-εj, 1i<j, εi-εj+kδ,ij 1i,j,k>0, kδ, k>0. These roots can be identified with pairs of integers (a,b), where a<b and 0b-1 via εa-εb= εamod- εb+kδ,where a=-(k+1) +iwith1i. The algebra U- has a PBW indexed by nonegative linear combinations of positive roots and these linear combinations correspond to the description of aperiodic multisegments as sums of indecomposable representations of the quiver.

2.2.4 By [Lus1998, Cor. 13.6] and [Lus1998, Cor 15.6] the canonical basis of U-=Uq(𝔰𝔩ˆ) is indexed by a subset of the aperiodic multisegments. On the other hand the PBW basis of U- is indexed by the set of aperiodic multisegments comparing dimensions (of graded components) allows one to conclude that the canonical basis is indexed by the set of aperiodic multisegments.

2.2.5 By [Lus1989-2, Cor. 10.7] the transition matrix between the canonical basis of the negative part of the quantum group Uq(𝔰𝔩ˆ) and the PBW basis is given by the Poincaré polynomials of the intersection cohomology sheaves for the quiver variety.

If c=α>0cααQ+ and let Vc be the corresponding sum of indecomposable representations of the quiver and let 𝕆c denote the corresponding GV orbit in EV=Ed. Let Eic=α>0 Eα(cα), where Eα(k)=Ti1TipEαip+1(k), Ei(k)=Eαik/[k]! and the order of the product and the definition of the Eα(k) are with respect to a fixed choice of reduced decomposition of w0.

[Lus1989-2, Cor. 10.7(a)] Let 𝕆c be an orbit in Ed. Let IC(μ*𝒞c) be the intersection cohomology complex of the closure of 𝕆c. Then bic=c Pc,cEic, where Pc,c=vd(c)-d(c)adim(H2a(IC(if!μ*𝒞c))) for f𝕆c.

CHECK THIS WITH LUSZTIG'S BOOK. Alternatively, [Lus1998, 2.2] we may define Li,a=μ! 𝒞i,a

2.2.6 Theorem ??? shows that these polynomials are Kazhdan-Lusztig polynomials for the affine symmetric groups Sn. The correspondence is made precise and summarized in the following theorem.

Let V be a Γ-graded vector space. Define Vj=tkVi for j=k+i with 1i, and fix the sequence of lattices L1L tL1,given byLi =j>iVj. Let x be a nilpotent representation of the quiver on the graded vector space V. Define the flag M1MtM1 by Mi= { j>ixji (vj)-vj| vjVj } This map has image in the subvariety of the affine flag variety given by Z= { M1Mt M1|Mi Li,dim(Li/Mi) =di,MiVi =Li } .

2.2.7 Define the canonical basis of Uv(𝔰𝔩ˆ) here.

2.2.8

(a) Let λ and μ be multisegments and let Eλ and Bμ be the corresponding elements of the PBW-basis and the canonical basis of Uv-(𝔤𝔩), respectively. Then Eλ=μ Pwλ,wμ (t)Bμ, where Py,x are the Kazhdan-Lusztig poylnomials of the symmetric group Sn, n=|λ|.
(b) Let λ and μ be aperiodic multisegments and let Eλ and Bμ be the corresponding elements of the PBW-basis and the canonical basis of Uv-(𝔰𝔩ˆ), respectively. Then Eλ=μ Pwλ,wμ (t)Bμ, where Py,x are the Kazhdan-Lusztig poylnomials of the affine Weyl group Sn, n=|λ|.

2.2.9 By [Lus1998, Cor. 11.8] the canonical basis "descends" to the modules L(Λ).

Canonical basis inside the Fock space

Assume that q is an th root of unity and Λ=Λa1++Λar where u1=q2a1,,ur=q2ar are the parameters of Hr,1,n. The generalized Fock space (Λ) is the Uv(𝔰𝔩ˆ) module given by (Λ)= λ=(λ(1),,λ(r)) (v)λ, with basis indexed by multipartitions with r components and with Uv(𝔰𝔩ˆ) action Khiλ = v|λ+|i-|λ-|i λ, Eiλ = c(λ/μ)=i v|λ-|i,>(λ/μ)-|λ+|i,>(λ/μ) μ, Kdλ = v-CΛ(λ)λ, Fiλ = c(μ/λ)=i v|λ+|i,<(μ/λ)-|λ-|i,<(μ/λ) μ, where |λ+|i = (# addable boxes of contentiofλ) |λ+|i,>b = (# addable boxes of contentiofλaboveb) |λ+|i,<b = (# addable boxes of contentiofλbelowb) A |λ-|i = (# removable boxes of contentiofλ) |λ-|i,>b = (# removable boxes of contentiofλaboveb) |λ-|i,<b = (# removable boxes of contentiofλbelowb) A Cu(λ) = i (# of boxes of contentuiofλ(i)) The irreducible highest weight module L(Λ) is the submodule of (Λ) generated by the highest weight vector (of weight Λ), L(Λ)=Uv (𝔰𝔩ˆ). Define a -linear involution on L(Λ) by v=v-1, Kh=K-h, Ei=Ei, Fi=Fi, andu =u, for uUv(𝔰𝔩ˆ). The canonical basis Bμ of L(Λ) is determined by the conditions Bμ=Bμ, andBμ=μ+ λ>μbλμ (v)λ,with bλμv[v].

The canonical basis of L(Λ) corresponds to the simple modules of the algebras Hr,1,n where the parameters in these algebras are determined by Λ. The Verma module M(Λ) has canonical basis in correspondence with the canonical basis of U- since M(Λ)U-vΛ. The canonical basis of M(Λ) corresponds to the simple modules of the affine Hecke algebras Hn and the PBW basis corresponds to the standard modules of Hn. Viewing L(Λ) as a quotient M(Λ)/I corresponds to restricting from Hn modules to Hr,1,n-modules. Under this restriction the PBW basis corresponds to the Specht modules of Hr,1,n. This gives an indexing of the Specht modules by aperiodic multisegments, and this corresponds (CHECK THIS) to the Jimbo indexing of the canonical basis of L(Λ).

Let λ=(λ(1),,λ(r)) be a multipartition. In λ, successively pair each unpaired addable node of content c with the nearest higher unpaired removable node of content c (if it exists) until there are no more possible pairings. If it exists, a good node of content c is the highest unpaired node of content c after pairing the addable and removable nodes of content c. Highest means that the node has maximal content among unpaired nodes in the partition λ(i) with i minimal. It would be best to define an ordering on the boxes??? A Kleshchev multipartition λ=(λ(1),,λ(r)) is an r-tuple of partitions such that one obtains =(,,) after consecutively removing good nodes. This definition is obtained by thinking of the crystal graph of L(Λ) as L(Λ)= L(Λi1) L(Λir).

The canonical basis Bμ of L(Λ) is indexed by the Kleshchev multipartitions μ=(μ(1),,μ(r)).

Let u1=qa1,,ur=qar be the parameters of the Ariki-Koike algebra Hr,1,n. Assume that the (a1,a2,,ar) are integers (we can always reduce to this case).

A cylindrical multipartition is a multipartition λ=(λ(1),,λ(r)) such that for each 1sr, λ(s)λ(s+1), as configurations on the same page. A cylindrical multipartition is -restricted if it is an aperiodic multisegment.

In this case the top element of the crystal graph is =(,,). The rule for adding boxes is the same as in the Kleshchev case except that order of reading the addable and removable nodes is different.

The canonical basis of L(Λ) is indexed by the cylindrical multisemgments.

The isomorphism L(Λ)L(ωi1)++L(ωik) gives the correpondence between the Jimbo parametrization by aperiodic multisegments and the parametrization of the nodes of the crystal graph by Kleshchev multipartitions.

Goodman and Wenzl

Goodman and Wenzl prove "directly" that the polynomials which describe the canonical basis are the same as parabolic KL polynomials by doing the appropriate matching up and then showing that they are both computed by the same algorithm. The algorithm is essentially that given by adding a single box at a time (applying a sequence of i induction functors) and computing the canonical basis inductively. The matching up is given as follows.

Consider the Fock space module L(Λ0) for the quantum group Uv(𝔰𝔩ˆ). The canonical basis of this module is indexed by regular partitions μ. The truncated Fock space has basis labeled by the regular partitions μ with (μ)k. Identify these with dominant integral weights of 𝔤𝔩k by letting λ=i=1k (λi-λi+1) ωi,whereωi= ε1++εi. ρ=(k-1)ε1+ (k-2)ε2++ εk-1+0εk. Associate partitions to weights of 𝔰𝔩k by λ=i=1k-1 (λi-λi+1)ωi

The affine Weyl group Sk of type Ak-1 is the reflection group defined by the hyperplane arrangement Hα+jδ= {x𝔥*|x,α=j} ,α>0,j. in the space 𝔥*=span{ε1,,εk}. Let H and H be the affine and Iwahori-Hecke algebras corresponding to Sk and Sk, respectively. Let 𝒜+ be the set of alcoves in the dominant chamber and consider the module IndHH(sgn)HH with basis Sk\Sk so that the action is given by NACs= { NAs+vNA, ifAs𝒜+ andAs>A, NAs+v-1 NA, ifAs𝒜+ andAs<A, 0, ifAs𝒜+. where the action of xSn on alcoves {wA+|wSn} is given by Ax=(wA+)x=wxA+.

The elements of Sn are tλw, λn and wSn. The nonextended affine Weyl group is { tλw|wSn, λ1+λk=0 } Thus Sk\Sk has coset representatives tλ, λn. We need to identify these elements with alcoves. Note that tλ acts as translation by λ1ε1++λkεk and w permutes the coordinates.

If λP+ then λ=(tλ1w) λ0,whereλ1 QP+andλ0 A+. Note that translating λ by -λ1 with λQP+ corresponds to removing boxes from a row of and placing them on a lower row. Since λ1P+ we must remove more boxes from the first row than the second, more from the second than the third, etc. The resulting sequence should be in the "fundamental hexagon" i.e. |λi-λi+1|. Then a permutation of this will put the result in the fundamental alcove and yield λ0.

A partition is -regular if it has no more than -1 rows of any given length. (The conjugate condition, that λ has no more than -1 columns of any given length, is called p-restricted, and says that λ is in the compact torus i(/)ωi.)

[GWe1999] Let nB,A(b) be the parabolic Kazhdan-Lusztig polynomials for H coming from the sign representation of H as given in ????. Let λ be a partition??? and μ an regular partition. Then [Sλ:Dμ]t= { na(λ+ρ),a(μ+ρ) (t), ifλis in the Skorbit of μ, 0, otherwise, and a(λ+ρ) is the alcove which contains λ+ρ in its closure and which lies on the positive side of all the hyperplanes containing μ.

Question: In what sense is the p-core of a partition the fundamental alcove representative of the W orbit of λ under the dot action?

Affine Hecke algebras

Following [Lus1998, 1.4] define the affine Hecke algebra H as the algebra given by generators T1,,Tn, and Xλ, λP, with relations TiTjTimij=TjTiTjmij, where π/mij is the angle between Hαi and Hαj,
Ti2=(q-q-1)Ti+1, 1in,
XλXμ=Xλ+μ, for λ,μP,
TiXλ=XsiλTi+(q-q-)Xλ-Xsiλ1-X-αi, for λP, 1in.

[Gro1994-2], [CGi1433132, Remark 8.8.8] Assume q1. The irreducible H modules are indexed by triples (s,x,χ) such that Z𝔤(y)𝔤-(s,q)𝒩 where (x,y,log(s)) is an 𝔰𝔩2 triple, and χ appears in H*(s,x) with nonzero multiplicity.

Using the map π:𝒩(s,q) given by projection onto the second factor, Mx=μ-1(n) is identified withs,x ={𝔟|Ad(s)𝔟=𝔟,x𝔟}, for all n𝒩(s,q) [CGi1433132, 8.1.7]. Then s,n,χ=H* (s,n)χ H-*(ix!μ*𝒞𝒩(s,q))χ ands,n,χ* =H*(s,n)χ H*(ix*μ*𝒞𝒩(s,q))χ are the standard and costandard modules for H*(Z(s,q)), respectively [CGi1433132, Prop. 8.6.15, 8.1.9, 8.1.11].

Decomposition numbers

The involutive antiautomorphism of H induced by switching the two factors of the Steinberg variety 𝒵=𝒩×𝒩𝒩 is given by Ti*=Ti, and(Xλ)* =Xλ,λP, and the bilinear form ,:Mx+tδ×Mx+tδ[t] is contravariant hm1,m2= m1,h*m2, form1,m2 x+tδ,hH. [Lus1998, Lemma 12.12].

The general method of ??? produces a Jantzen filtration s,x,χ= s,x,χ(0) s,x,χ(1) of s,x,χ and the generalized decomposition numbers for the affine Hecke algebra H(q) are the polynomials [s,x,χ:Ls,x,χ]t =k [ s,x,χ(k)/ s,x,χ(k+1) :Ls,x,χ ] tk. (4.3) Theorem ??? applied to this situation yields the following refinement of [CGi1433132, Theorem 8.6.23].

[Gin1987-2, Theorem 2.6.2] and [Gro1994-2]. [s,x,χ:Ls,x,χ]t =ktkdimHk (ix!ICx,χ).

Type A affine Hecke algebras

The affine Hecke algebra of type A is the algebra H given by generators Xε1,T1,,Tn-1 and relations

(a) TiTi+1Ti=Ti+1TiTi+1, for 1in-2,
(b) TiTj=TjTi, if |i-j|>1,
(c) Ti2=(q-q-1)Ti+1, for 1in-1,
(d) Xε1Ti=TiXε1, if i>1,
(e) Xε1T1Xε1T1=T1Xε1T1Xε1.
Let L=i=1nεi ,andXεi= Ti-1T2T1Xε1 T1T2Ti-1, for1in, and define Xλ=(Xε1)λ1(Xεn)λn, for λ=λ1ε1++λnεn in L. If w is in the symmetric group Sn define Tw=Ti1Tip for a reduced expression w=si1sip of w in terms of the simple reflections si=(i,i+1), 1in-1.

[Reference???]

(a) The elements Xε1,,Xεn all commute with each other.
(b) The elements XλTw, λL, wSn, are a basis of H.
(c) The center Z(H) is the ring of symmetric functions [X±ε1,,X±εn]Sn.

An H module M has central character s=(s1,,sn)(*)n if p(Xε1,,Xεn) m=p(s1,,sn)m, for allp[X±ε1,,X±εn]Sn =Z(H). The central character of a module M is unique up to the Sn action permuting the coordinates of s.

The aperiodic multisegments with n boxes index the irreducible representations of the affine Hecke algebra Hn of type A.

Let λ1,,λm be the sizes of the Jordan blocks of x and let rj=λ1++λj. Let Hλ be the "parabolic" subalgebra of H generated by Xε1,,Xεn and Ti, i{r1,,rm}. Define a one dimensional representation s,x=vs,x of Hλ by Xεkvs,x=sk vs,x,1kn, andTivs,x= qvs,x,for i{r1,,rm}. Then the standard module s,x of H is given by s,xH Hλs,x.

From Theorem ??? we obtain the following corollary (a refinement of [Zel1985, Cor. 1]). An alternative way of obtaining this result is explained in [OR].

Let H be the affine Hecke algebra corresponding to GLn().

(a) If q is not a root of 1, the generalized decomposition numbers for the affine Hecke algebra are [Ms,x:Ls,x]t =Pwx,wx(t), where Pu,w(t) are the Kazhdan-Lusztig polynomials of the symmetric group Sn.
(2) If q is a root of unity the generalized decomposition numbers for the affine Hecke algebra are [Ms,x:Ls,x]t =Pwx,wx(t), where Pu,w(t) are the Kazhdan-Lusztig polynomials of the affine symmetric group Sn.

Put in a converse to this last theorem.

Cyclotomic Hecke algebras

Let 𝔽 be a field, u1,,ur𝔽 and q𝔽*. The cyclotomic Hecke algebra Hr,1,n(u1,,ur;q) is the affine Hecke algebra with the additional relation (Xε1-u1) (Xε2-u2) (Xε1-ur)=0. Note that Hr,1,n(u1,,ur;q)=Hr,1,n(uπ(1),,uπ(r);q) for any permutation π.

Indexing the simple modules

A partition λ=(λλ2) is p-restricted if λi-λi+1<p for all i1.

[Mat1998, Theorem 4.8 and Theorem 3.7] Let p=char(𝔽). Let u1,,urk and qk*, and consider the cyclotomic Hecke algebra Hr,1,n with these parameters.

(a) Assume q1, u1,,uk are nonzero and uk+1==ur=0. Then the simple Hr,1,n modules Dμ are indexed by multipartitions μ=(μ(1),,μ(r)) such that
(1) μ=(μ(1),,μ(k)) is a Kleshchev multipartition,
(2) μ(k+1)==μ(k-1)=
(3) μ(r) is p restricted.
(b) If q=1 the simple Hr,1,n-modules Dμ are indexed by multipartitions μ=(μ(1),,μ(r)) such that
(1) Each μ(i), 1ir, is p restricted,
(2) μ(i)= if there exists j>i such that uj=ui.

Decomposition numbers

[Ar2] The algebra Hr,1,n is semisimple if and only if

(a) uiujq2k for ij, k, |k|<n,
(b) q2 is not a primitive th root of unity for any 2n.

In the generic case, when the algebra Hr,1,n=Hr,1,n(u1,,ur;q) is semisimple the irreducible representations Sλ are indexed by r-tuples λ=(λ(0),,λ(r)) of partitions with n boxes total. One can construct an integral form 𝒜Hr,1,n of the Ariki-Koike algebra and an integral form 𝒜Sλ of the module Sλ so that the module Sλ=𝔽𝒜 𝒜Sλ, can be defined for the algebra Hr,1,n for any choice of u1,,ur𝔽, q𝔽*. This module is called the Specht module for the algebra Hr,1,n (though it ought to be called a Young module).

There is a unique Hr,1,n contravariant form on Sλ, ,t: Stλ×Stλ [t]such that vλ,vλ =1, where vλ is the vector indexed by the row reading tableau of shape λ. Letting Sλ(j)= { mStλ| m,n tj[t],for all nStλ } , the Jantzen filtration of Sλ is Sλ=(Sλ)(0) (Sλ)(1) where(Sλ)(j) =Sλ(j)tSλ(j).

The generalized decomposition numbers of the Ariki-Koike algebras are the polynomials [Sλ:Dμ]t= k [(Sλ)(k)(Sλ)(k+1):Dμ] tk. (3.3)

The generalized decomposition number for the Ariki-Koike algebra are given by KL polynomials. [Sλ:Dμ]t= n??(t), where nBA(t) is the parabolic Kazhdan-Lusztig polynomial of the affine Hecke algebra corresponding to the sign representation of H.

Iwahori-Hecke algebras

The Iwahori-Hecke algebra of type A is the algebra Hn(q)=H1,1,n. From Theorems ??? and ???

(a) Hn(q) is semisimple if and only if q2 is not a primitive th root of 1 for 2n.
(b) The simple Hn(q) modules Dμ are indexed by the regular partitions with n boxes.

A partition μ is -regular if it is an aperiodic multisegment. The irreducible representations Dμ of the Iwahori-Hecke algebra Hn(q) where q is a primitive th root of unity are indexed by the regular partitions μ with n boxes. The Specht modules Sλ are indexed by the partitions λ of n.

(LLT conjecture) Let dλμ(t) be the coefficients in the expansion of the canonical basis of the Fock space in terms of the PBW basis. If λ is a partition and μ is an regular partition then dλμ(t)= [Sλ:Dμ]t, where Sλ is the Specht module and Dμ is the irreducible module for the Iwahori-Hecke algebra of type A when q is a primitive th root of unity.

Connection to the quantum group

The proof of the following theorem is a "direct calculation" which verifies that the i-induction and i-restriction functors act on Specht modules in the same way that the Ei and Fi operators act on the multipartition basis of the Fock space.

The algebra U- is generated by the fi, i/, with Serre relations (WHICH ARE???).

[Ari1996] The i-induction and i-restriction functors make the direct sum of the Grothendieck groups of the categories of finite dimensional modules of the Ariki-Koike algebras Hr,1,n(qa1,,qar), n0, into an irreducible highest weight Uv(𝔰𝔩ˆ) module of highest weight Λ=Λa1++Λar, where Λi are the fundamental weights of Uv(𝔰𝔩ˆ).

Proof.

According to [Ari1996, Lemma 4.2] Fs,xFi= cs,xsx Fs,x, where cs,xs,x= (# ways to add a box of contentiat the end of a row of xand getx.

A similar calculation verifies that the i-induction and i-restriction functors act on standard modules in the same way that the Ei and Fi operators act on the PBW basis of U- for the quantum group of type A or type A-1.

It is interesting to note that every standard module for Hn is a Specht module for some choice of parameters for Hr,1,n.

[Ari1996] The i-induction and i-restriction functors make the direct sum of the Grothendieck groups of the categories of finite dimensional modules of the affine Hecke algebras Hn of type A into a Uv(𝔰𝔩ˆ) module isomorphic to U-.

Under this identification the simple modules form the canonical basis of L(Λ) and the standard modules form the "PBW" basis. There is an inner product on the quantum group which should "become" the inner product/Jantzen filtrations on modules for the affine Hecke algebra.

Kac-Moody Lie algebras

Let 𝔤 be a -graded Lie algebra such that 𝔤0 is reductive and 𝔤 is semisimple for ad𝔤0.

Let 𝒪 be the category of graded 𝔤 modules which are locally finite for 𝔟=𝔤0 and semisimple for 𝔤0. Let Λ be set of isomorphism classes of irreducible finite dimensional -graded 𝔤0. modules and for EΛ define Δ(E)=U𝔤U𝔟 E,and(E) =Homg0(U𝔤,E), A tilting module is a module T which admits a Δ flag and a flag. Define L(E) = the unique simple quotient (i.e. the head) ofΔ(E). = the unique simple submodule (i.e. the socle) of(E) ,and T(E) = the tilting module with parameterE, where the tilting module with parameter E is the unique indecomposable module such that

(a) Ext𝒪1(Δ(F),T)=0 for all FΛ,
(b) T(E) has a Δ-flag 0=T0T1 with T1=Δ(E).

Suppose, in addition, that 𝔤0 contains an element such that [,X]=iX for all i and X𝔤i. Let 𝒪 be the category of all 𝔤-modules which are locally finite for 𝔟 and semisimple for 𝔤0. This is the category 𝒪 "without the grading". Let Λ be the set of isomorphism classes of finite dimensional irreducible representations of 𝔤0.

If 𝔤 is a Kac-Moody algebra then 𝔤0=𝔥, 𝔤1=i=0n 𝔤αiand Λ=𝔥*. The Kazhdan-Lusztig conjecture for the case when 𝔤 is symmetrizable was established by Kashiwara.

[Kas90] Assume 𝔤 is symmetrizable and let λΛ=𝔥* be such that λ+ρ,α>0. Then [Δ(xλ):L(yλ)] =Px,y(1), where Px,y(t) is the Kazhdan-Lusztig polynomial for the Weyl group corresponding to 𝔤.

Let J be a subset of the simple roots and let 𝔤J be the corresponding "parabolic" subalgebra of 𝔤. Let ΔJ(λ)=Δ(λ) for λ a dominant integral weight for 𝔤J.

[Soe1998, Prop. 7.5] Let 𝔤 be symmetrizable. Let λ𝔥* be such that λ,α0 for all simple roots α. Then [ΛJ(xλ):L(ν)]= { nx,y(1), ifν=yλ withyWJ, 0, otherwise.

Proof.

Let L(λ) be the finite dimensional irreducible representation of 𝔤J of highest weight λ. Let M(μ) be the Verma module for 𝔤J of highest weight μ. Applying the functor U𝔤U𝔟 to the BGG-resolution 0𝒞(wJ) 𝒞1𝒞0L (λ)0,where 𝒞k=wWJ(w)=k M(wλ), gives an exact sequence 0U𝔤U𝔟𝒞(w0) U𝔤U𝔟𝒞1 U𝔤U𝔟𝒞0 Δ(λ)0, for λ a dominant integral weight for 𝔤J. Thus, for all ν𝔥* and λ𝔥* such that λ,αi0 for αiJ [Δ(λ):L(ν)] =wW(-1)(w) [Δ(wλ):L(ν)], The result now follows from ??? and ???.

Quantum groups

This introductory material is taken from [CP, § 11.2].

Let ξ be a primitive th root of unity where is odd and greater than di for all i. All representations are complex representations.

Let Vq(λ) be the irreducible Uq-module with highest weight λP and highest weight vector vλ. Let V𝒜(λ) be the U𝒜res-module generated by vλ and define the Δξres(λ)= V𝒜res(λ)𝒜 ,Weyl module, via the homomorphism 𝒜 given by qξ. Define Vξres(λ)= the unique simple quotient ofWξres(λ).

[CP, Theorem 11.2.8] The finite dimensional simple Uξres-modules of type 1 are Vξres(λ), λP+.

Let F:UξresU be the Frobenius map. Let V(λ) denote the simple U-module of highest weight λ let F*(V(λ)) be its pullback.

[CP, 11.2.9 and 11.2.11] (Tensor product theorem) Let λP+ and write λ=λ0+λ1 with λ0,λ1P+, and λ0,αi< for 1in. Then Vξres(λ) Vξres(λ0) F*(V(λ)), andV(λ)F Vξres(λ).

The following theorem was conjectured by Lusztig [Lu9?] and proved by Kazhdan-Lusztig via the passage to the category of representations of affine Lie algebras of level -h- via Theorem ??? below.

Assume that 𝔤 is type A, D or E and let θ be the highest root. Let A-= { λ𝔥*| λ+ρ,θ -, λ,αi -1,1in } be the chamber bounded by the hyperplanes λ+ρ,θ =-,λ,αi =-1,1in, and let W be the corresponding affine Weyl group. Let wλ be of minimal length such that wλ-1λA-.

Let λP+. Then [ Wξres(wλ) :Vξres(λ) ] =(-1)(wwλ) Pw,wλ(1), where wλW is of minimal length such that wλ-1λ is in A+ and Py,x is the Kazhdan-Lusztig polynomial for the affine Weyl group W.

[Soe1998, §7] explains that (at least within the Grothendieck ring of the principal block of the category of finite dimensional Uξ modules) this formula can be inverted and written in the form AnB,Aˆ (1)LA=C, The principal block is the smallest direct summand of the category of finite dimensional Uξ modules which contains the trivial representation. Here LA denotes the simple and C is the module Wξres(λ).

Let 𝔤 be a finite dimensional complex semisimple Lie algebra. Let 𝔤=𝔤 [t,t-1], 𝔤=𝔤z, and𝔤=𝔤 . Then 𝔤 is the loop algebra and 𝔤 is an affine Kac-Moody Lie algebra.

[KLu1980, I-IV] For many -0 there is an equivalence of categories 𝒪e(K=-h-) Uζ-mode,1. where 𝒪e(K=-h-) is the category of 𝔤 modules M of finite length such that

(a) K=2hz acts by the value -h-,
(b) 𝔤0 acts locally finitely, and
(c) 𝔤0 acts locally nilpotently,
and Uζ-mode,1 is the category of finite dimensional type 1 representations of the quantum group Uq(𝔤) where q=e-iπ/.

Let be the principal block (the smallest direct summand containing the trivial representation) of the category of finite dimensional Uζ(𝔤) modules where ζ is a primitive th root of unity. The simple objects LA of are indexed by the set 𝒜+ of alcoves in the dominant chamber. By the Kazhdan-Lusztig equivalence the block is equivalent to the block of 𝒪 containing L(μ) with μ=-(h+)2h γ,whereγ=2ρ-2 ρf, where ρ𝔥* is such that ρ,αi=1 for all 0in, and ρf is the half sum of the positive roots of 𝔤0=𝔤z.

[Soergel] Let W be the affine Weyl group of the Weyl group W of 𝔤 and let H and H be the corresponding affine and Iwahori-Hecke algebras, respectively. Let TA be the unique indecomposable tilting module in which admits a flag ending with a surjection TAA. Then [TA:B]= nB,A(1), where nB,A(v) is the parabolic Kazhdan-Lusztig polynomial for H corresponding to the sign representation of H.

Algebraic groups in characteristic p

Let 𝔽 be a field of characteristic p and let G be a semisimple split simply connected algebraic group over 𝔽. Let B be a Borel subgroup of G and T a maximal torus.

Let λX(T). The Weyl module associated to λ is H0(λ)=H0 (G/B,λ)= IndBG𝔽λ. The simple module are L(λ) where L(λ) is the unique simple submodule of H0(λ).

[Soe2000, Theorem 1.2] Let ix!ICy be the stalk of the intersection homology sheaf on the Schubert variety Xw at a point of the Schubert cell Xx. Then [ H0(st+ρ):L (st+yρ) ] =idimk (Hk(ix!ICy)).

This is part of the Lusztig conjecture [Lu7].

Let K be an algebraically closed field of characteristic p. Let G be a reductive affine algebraic group and let B a Borel subgroup and TB a maximal torus. Let W be the Weyl group, X=X(T) the weight lattice and X+ the set of dominant integral weights. If λX Let (λ) = IndBG(Kλ), Δ(λ) = (-w0λ)* =the Weyl module of highest weightλ, L(λ) = soc((λ)) =head(Δ(λ)). T(λ) = indecomposable tilting module of highest weightλ, P(λ) = the projective cover ofL(λ),

The simple G-modules are {L(λ)|λX+}.

If (p-1)ρX+ The Steinberg module is ((p-1)ρ) =L((p-1)ρ) =Δ((p-1)ρ) =T((p-1)ρ). Let F:GG be a Frobenius map.

(Steinberg tensor product theorem) [Jz, II (3.19)] ((p-1)ρ) (μ)F ((p-1)ρ+pμ).

[Don1993, Prop. 2.1] Let μX+ and let λX+ be such that λαip-1, for 1in. Then T(2(p-1)ρ+w0λ) T(μ)FT (2(p-1)ρ+w0λ+pμ).

The general linear group

Let K be an algebraically closed field of characteristic p.

Let GLn(K) be the general linear group with entries in K. Let Δ(λ) = Weyl module with highest weightλ, (λ) = the contravariant dual ofΔ(λ), T(λ) = the indecomposable tilting module with highest weightλ, L(λ) = simpleGLn(K) -module of highest weightλ. The modules Δ(λ) are the reductions mod p of the characteristic 0 irreducible representations of GLn(). The L(λ) are the simple GLn(K) modules. The simple module L(λ) is the unique module in the head of Δ(λ) and the unique module in the socle of (λ). The tilting module T(λ) is the unique indecomposable module of highest weight λ which has both a Δ and filtration.

(a) [T(λ):Δ(μ)]=[Δ(μ):L(λ)]. for arbitrary heighest weights, what does conjugation mean in this context?
(b) (Steinberg's tensor product theorem) Δ(μ)FΔ((p-1)δ)Δ(pμ+(p-1)δ).
(c) [Er, Lemma 2.1b] T(μ)FΔ((p-1)δ)T(pμ+(p-1)δ).

Proof.

(c) Note that T((p-1)ρ)Δ((p-1)ρ)L((p-1)ρ) and apply Donkin's result [Don1993, Proposition 2.1]

The symmetric group

Let K be an algebraically closed field of characteristic p. Let Sr be the symmetric group.

The irreducible representations of the symmetric group in characterstic 0 are indexed by the partitions μ with r boxes. Their reductions mod p are the Specht modules Sμ.

A p-regular partition is a partition μ that does not have p equal parts.

The irreducible KSr modules Dμ are indexed by the p regular partitions μ with r boxes.

[Gre1980, Theorem 6.6g] and [Er, Proposition 2.3]

(a) Let λ,μr, (λ)n, (μ)n, such that λ is p regular. [T(λ):Δ(μ)]= [Sμ:Dλ].
(b) Let λ and μ be partitions of r with length n. Then [Δ(μ):L(λ)]= [Spμ+(p-1)δ:Dpλ+(p-1)δ].

Proof.

(a) Apply the Schur functor. (b) is a consequence of the following equality. [Δ(μ):L(λ)] = [T(λ),Δ(μ)] = [ T(pλ+(p-1)δ): Δ(pμ+(p-1)δ) ] = [ Spμ+(p-1)δ: Dpλ+(p-1)δ ] .

References

[Ari1996] S. Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), J. Math. Kyoto Univ. 36 (1996), 789–808.

[AMa2000] S. Ariki and A. Mathas, On the number of simple modules of the Hecke algebras of type G(r,p,n), Math. Zeitschrift 233 (2000), no. 3, 601–623.

[Don1993] S. Donkin, On tilting modules for algebraic groups, Math. Zeit. 212 (1993), 39-60.

[Erd1996????] K. Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316-320.

[FLO1999] O. Foda, B. Leclerc, M. Okado, J.-Y. Thibon, T. Welsh, Branching functions of An-1(1) and Jantzen-Seitz problem for Ariki-Koike algebras, Adv. Math. 141 (1999), 322-365.

[GWe1999] F. Goodman and H. Wenzl, Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials, Int. Math. Res. Not. 5 (1999), 251-275.

[Gro1994-2] I. Grojnowski, Representations of affine Hecke algebras (and affine quantum GLn) at roots of unity International Math. Res. Notices 5 (1994), 215-217.

[Gro1996] I. Grojnowski, Jantzen filtrations, unpublished notes, 1996.

[Gin1987-2] V. Ginzburg, Geometrical aspects of representation theory, Proc. Int. Cong. Math. (Berkeley 1986), Vol 1, Amer. Math. Soc. 1987, 840-848.

[Gre1980] J.A. Green, Polynomial representations of GLn, Lecture Notes in Mathematics 830, Springer-Verlag, New York, 1980.

[JMM1991] M. Jimbo, K. Misra, T. Miwa, M. Okado, Combinatorics of representations of Uq(𝔰𝔩ˆ(n)) at q=0, Comm. Math. Phys. 36 (1991), 543-566.

[KLu1980] D. Kazhdan and G. Lusztig, Schubert Varieties and Poincare Duality, Proc. Symp. Pure Math. 36 (1980), 185-203. MR84g:14054

[KSc1985] M. Kashiwara and P. Schapira, Microlocal study of sheaves, Astérisque 128 (1985).

[LLT1996] A. Lascoux, B. Leclerc, J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205-263.

[Lus1981] G. Lusztig, Green polynomials and singularities of nilpotent classes, Adv. in Math. 42 (1981), 169-178. MR83c:20059

[Lus1990] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc 3 (1990), 447-498.

[Lus1991] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc 4 (1991), 365-421.

[Lus1989-2] G. Lusztig, Representations of affine Hecke algebras, in Orbites unipotentes et représentations, Astérisque 171-172 (1989), 73-89.

[Lus1998] G. Lusztig, Bases in equivariant K-theory, Rep. Theory 2 (1998), 298-369.

[Lus1987-3] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Symp. Pure Math. 37 Amer. Math. Soc. (1987), 313-317.

[Mat1998] A. Mathas, Simple modules of Ariki-Koike algebras, Proc. Symp. Pure Math. Amer. Math. Soc. 63 (1998), 383-396.

[Soe1998] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Representation Theory 0 (1998), 432-448.

[Soe2000] W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, Journal of Pure and Applied Algebra 152 (2000), 311-335.

[Zel1985] A. Zelevinsky, Two remarks on graded nilpotent classes, Russ. Math. Surv. 40 (1985), 249-250.

Notes and References

These notes are from /Work2004/Dell_Laptop/Unpublished/Decompnos/conndecomp8.11.00.tex

Research supported in part by National Science Foundation grant DMS-9622985.

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