Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 26 March 2015

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.


Hecke algebras

This section gives some of the main representation theoretic results used in this thesis, and defines Hecke algebras. Three roughly equivalent structures are commonly used to describe the representation theory of an algebra A: (1) Modules. Modules are vector spaces on which A acts by linear transformations. (2) Representations. Every choice of basis in an n-dimensional vector space on which A acts gives a representation, or an algebra homomorphism from A to the algebra of n×n matrices. (3) Characters. The trace of a matrix is the sum of its diagonal entries. A character is the composition of the algebra homomorphism of (2) with the trace map. The following discussion will focus on modules and characters.

Modules and characters

Let A be a finite dimensional algebra over the complex numbers . An A-module V is a finite dimensional vector space over with a map A×V V (a,v) av such that (c1)v = cv, c, (ab)v = a(bv), a,bA,vV, (a+b) (c1v1+c2v2) = c1av1+ c2av2+ c1bv1+ c2bv2, c1,c2. An A-module V is irreducible if it contains no nontrivial, proper subspace V such that avV for all aA and vV.

An A-module homomorphism θ:VV is a -linear transformation such that aθ(v)=θ(av), foraA,vV, and an A-module isomorphism is a bijective A-module homomorphism. Write HomA(V,V) = {A-module homomorphismsVV} EndA(V) = HomA(V,V).

Let Aˆ be an indexing set for the irreducible modules of A (up to isomorphism), and fix a set {Aγ}γAˆ of isomorphism class representatives. An algebra A is semisimple if every A-module V decomposes VγAˆ mγ(V)Aγ, wheremγ(V) 0,mγ (V)Aγ= AγAγmγ(V)terms. In this thesis, all algebras (though not necessarily all Lie algebras) are semisimple.

(Schur's Lemma). Suppose γ,μAˆ with corresponding irreducible modules Aγ and Aμ. Then dim(HomA(Aγ,Aμ))= δγμ= { 1, ifγ=μ, 0, otherwise.

Suppose AB is a subalgebra of B and let V be an A-module. Then BAV is the vector space over presented by generators {bv|bB,vV} with relations bav = bav, aA,bB,vV, (b1+b2) (v1+v2) = b1v1+ b1v2+ b2v1+ b2v2, b1,b2B, v1,v2V. Note that the map B×(BAV) BAV (b,bv) (bb)v makes BAV a B-module. Define induction and restriction (respectively) as the maps IndAB: {A-modules} {B-modules} V BAV ResAB: {B-modules} {A-modules} V V

Suppose V is an A-module. Every choice of basis {v1,v2,,vn} of V gives rise to an algebra homomorphism ρ:AMn(), whereMn()= {n×nmatrices with entries in}. The character χV:A associated to V is χV(a)= tr(ρ(a)). The character χV is independent of the choice of basis, and if VV are isomorphic A-modules, then χV=χV. A character χV is irreducible if V is irreducible. The degree of a character χV is χV(1)=dim(V), and if χ(1)=1, then a character is linear; note that linear characters are both irreducible characters and algebra homomorphisms.

Define R[A]=-span {χγ|γAˆ} ,whereχγ= χAγ, with a -bilinear inner product given by χγ,χμ =δγμ. (2.1) Note that the semisimplicity of A implies that any character is contained in the subspace 0-span {χγ|γAˆ}. If χV is a character of a subalgebra A of B, then let IndAB(χV)= χBAV and if χV is a character of B, then ResAB(χV) is the restriction of the map χV to A.

(Frobenius Reciprocity). Let A be a subalgebra of B. Suppose χ is a character of A and χ is a character of B. Then IndAB(χ),χ= χ,ResAB(χ).

Weight space decompositions

Suppose A is a commutative algebra. Then dim(Aγ)=1, for allγAˆ. The corresponding character χγ:A is an algebra homomorphism, and we may identify the label γ with the character χγ, so that av=χγ(a)v= γ(a)v,for aA,vAγ.

If A is a commutative subalgebra of B and V is an B-module, then as a A-module V=γAˆ Vγ,whereVγ ={vV|av=γ(a)v,aA}. The subspace Vγ is the γ-weight space of V, and if Vγ0 then V has a weight γ.

For large subalgebras A, such a decomposition can help construct the module V. For example, 1. If 0vγVγ, then {vγ}Vγ0 is a linearly independent set of vectors. In particular, if dim(Vγ)=1 for all Vγ0, then {vγ} is a basis for V. 2. If V is irreducible, then BVγ=V. Chapter 6 uses this idea to construct irreducible modules of the Yokonuma algebra.

Characters in a group algebra

Let G be a finite group. The group algebra G is the algebra G=-span{gG} with basis G and multiplication determined by the group multiplication in G. A G-module V is a vector space over with a map G×V V (g,v) gv such that (gg)v=g(gv), g(cv+cv)=cgv+cgv, forc,c,g,gG,v,vV. Note that there is a natural bijection {G-modules} {G-modules} V V , so I will use G-modules and G-modules interchangeably. Let Gˆ index the irreducible G-modules.

The inner product (2.1) on R[G] has an explicit expression ·,·: R[G]×R[G] (χ,χ) 1GgGχ(g)χ(g-1), and if U is a subgroup of G, then induction is given by IndUG(χ): G g 1UxGxgx-1Uχ(xgx-1).

Hecke algebras

An idempotent eG is a nonzero element that satisfies e2=e. For γGˆ, let eγ= χλ(1)G gGχγ (g-1)g G. These are the minimal central idempotents of G, and they satisfy 1=γGˆ eγ,eγ2= eγ,eγ eμ=δγμ eγ,Geγ dim(Gγ)Gγ. In particular, as a G-module, GγGˆ Geγ.

Let U be a subgroup of G with an irreducible U-module M. The Hecke algebra (G,U,M) is the centralizer algebra (G,U,M)= EndG (IndUG(M)). If eMU is an idempotent such that MUeM, then IndUG(M)= GU UeM=G UeM GeM and the map θM: eMGeM (G,U,M) eMgeM ϕg where ϕg: GeM GeM keM keMgeM, (2.2) is an algebra anti-isomorphism (i.e. θM(ab)=θM(b)θM(a)).

The following theorem connects the representation theory of G to the representation theory of (G,U,M). Theorem 2.3 and the Corollary are in [CRe1981, Theorem 11.25].

(Double Centralizer Theorem). Let V be a G-module and let =EndG(V). Then, as a G-module, VγGˆ dim(γ)Gγ if and only if, as an -module, VγGˆ dim(Gγ)γ where {γ|γGˆ,γ0} is the set of irreducible -modules.

Suppose U is a subgroup of G. Let MUeM be an irreducible U-module. Write =(G,U,M). Then the map {G-submodules ofGeM} {-modules} V eMV is a bijection that sends Gγγ for every irreducible Gγ that is isomorphic to a submodule of IndUG(M) (GγIndUG(M)).

Chevalley Groups

There are two common approaches used to define finite Chevalley groups. One strategy considers the subgroup of elements in an algebraic group fixed under a Frobenius map (see [Car1985]). The approach here, however, begins with a pair (𝔤,V) of a Lie algebra 𝔤 and a 𝔤-module V, and constructs the Chevalley group from a variant of the exponential map. This perspective gives an explicit construction of the elements, which will prove useful in the computations that follow (see also [Ste1967]).

Lie algebra set-up

A finite dimensional Lie algebra 𝔤 is a finite dimensional vector space over with a -bilinear map [·,·]: 𝔤×𝔤 𝔤 (X,Y) [X,Y] such that (a) [X,X]=0, for all X𝔤, (b) [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0, for X,Y,Z𝔤. Note that if A is any algebra, then the bracket [a,b]=ab-ba for a,bA gives a Lie algebra structure to A.

A 𝔤-module is a vector space V with a map 𝔤×V V (X,v) Xv such that (aX+bY)v = aXv+bYv, for alla,b, X,Y𝔤,vV, X(av1+bv2) = aXv1+ bXv2, for alla,b, X𝔤,v1,v2V, [X,Y]v = X(Yv)- Y(Xv), for allX,Y𝔤,vV. A Lie algebra 𝔤 acts diagonally on V if there exists a basis {v1,,vr} of V such that Xvivi, for allX𝔤and i=1,2,,r.

The center of 𝔤 is Z(𝔤)= {X𝔤|[X,Y]=0for allY𝔤}. Let γ𝔤ˆ index the irreducible 𝔤-modules 𝔤γ. A Lie algebra 𝔤 is reductive if for every finite-dimensional module V on which Z(𝔤) acts diagonally, V=γ𝔤ˆ mγ(V)𝔤γ, wheremγ(V) 0. If 𝔤 is reductive, then 𝔤=Z(𝔤)𝔤s where𝔤s=[𝔤,𝔤] is semisimple.

Let 𝔤 be a reductive Lie algebra. A Cartan subalgebra 𝔥 of 𝔤 is a subalgebra satisfying (a) 𝔥k=[𝔥k-1,𝔥]=0 for some k1 (𝔥 is nilpotent), (b) 𝔥={X𝔤|[X,H]𝔥,H𝔥} (𝔥 is its own normalizer). If 𝔥s is a Cartan subalgebra of 𝔤s, then 𝔥=Z(𝔤)𝔥s is a Cartan subalgebra of 𝔤. Let 𝔥*=Hom (𝔥,)and 𝔥s*= Hom(𝔥s,).

As an 𝔥s-module, 𝔤s decomposes 𝔤s𝔥s 0α𝔥s* (𝔤s)α,where (𝔤s)α= X𝔤s|[H,X]=α(H)X,H𝔥s. The set of roots of 𝔤s is R={α𝔥*|α0,(𝔤s)α0}. A set of simple roots is a subset {α1,α2,,α}R of the roots such that (a) 𝔥s*=-span{α1,α2,,α} with {α1,,α} linearly independent, (b) Every root βR can be written as β=c1α1+c2α2++cα with either {c1,,c}0 or {c1,,c}0. Every choice of simple roots splits the set of roots R into positive roots R+= {βR|β=c1α1+c2α2++cα,ci0} and negative roots R-= {βR|β=c1α1+c2α2++cα,ci0} . In fact, R-=-R+.

Example. Consider the Lie algebra 𝔤𝔩2=𝔤𝔩2()={2×2matrices with entries in}, and bracket [·,·] given by [X,Y]=XY-YX. Write 𝔤𝔩2=Z(𝔤𝔩2) 𝔰𝔩2, where Z(𝔤𝔩2)={(c00c)|c} and 𝔰𝔩2={X𝔤𝔩2|tr(X)=0}. Also, 𝔰𝔩2= {(a00-a)|a} {(00c0)|c} {(0c00)|c}. Let α𝔥*={(a00b)|a,b}* be the simple root given by α(a00b)=a-b, so that R+={α} and R-={-α}.

From a Chevalley basis of 𝔤s to a Chevalley group

For every pair of roots α,-α, there exists a Lie algebra isomorphism ϕa:𝔰𝔩2𝔤α,𝔤-α. Under this isomorphism, let Xα=ϕa (0100) (𝔤s)α, Hα=ϕα (100-1) 𝔥s, X-α=ϕα (0010) (𝔤s)-α. Note that [Xα,X-α]=Hα. The following classical result can be found in [Hum1972, Theorem 25.2].

(Chevalley Basis). There is a choice of the ϕα such that the set {Xα,Hαi|αR,1i} is a basis of 𝔤s satisfying (a) Hα is a -linear combination of Hα1,Hα2,,Hα. (b) Let α,βR such that β±α, and suppose l,r0 are maximal such {β-lα,,β-α,β,β+α,,β+rα} R. Then [Xα,Xβ]= { ±(l+1) Xα+β, ifr1, 0 otherwise.

A basis as in Theorem 2.5 is a Chevalley basis of 𝔤s (For an analysis of the choices involved see [Sam1969] and [Tit1966]).

Let V be a finite dimensional 𝔤-module such that V has a -basis {v1,v2,,vr} that satisfies (a) There exists a -basis {H1,,Hn} of 𝔥 such that (1) Hαi0-span{H1,,Hn}, (2) Hivjvj for all i=1,2,,n and j=1,2,,r. (3) dim(-span{H1,H2,,Hn})dim(𝔥). (b) Xαnn!vi-span{v1,v2,,vr} for αR, n0 and i=1,2,,r. (c) dim(-span{v1,v2,,vr})dim(V). (Condition (a) guarantees that Z(𝔤) acts diagonally. If Z(𝔤)=0, then the existence of such a basis is guaranteed by a theorem of Kostant [Hum1972, Theorem 27.1]).

Let V be a finite dimensional 𝔤-module that satisfies (a)-(c) above. For H𝔥, let (Hn)= (H-1)(H-2)(H-(n+1))n! End(V). As a transformation of the basis v1,v2,,vr, the element Hi=diag(h1,h2,,hr)End(V) with hj. Let y*. Temporarily abandon precision (i.e. allow infinite sums) to obtain n0 (y-1)n (Hin) = diag ( n0(y-1)n (h1n), n0(y-1)n (h2n),, n0(y-1)n (hrn), ) = diag(yh1,yh2,,yhr) End(V)(by the binomial theorem). This computation motivates some of the definitions below.

Let 𝔥=-span {H1,H2,,Hn}. (2.3) The finite field 𝔽q with q elements has a multiplicative group 𝔽q* and an additive group 𝔽q+. Let Vq=𝔽q-span {v1,v2,,vr}. (2.4)

The finite reductive Chevalley group GVGL(Vq) is GV= xα(a),hH(b) |αR,H𝔥,α 𝔽q,b𝔽q* , where xα(a) = n0 anXαnn!, and (2.5) hH(b) = diag(bλ1(H),bλ2(H),,bλr(H)), whereHvi=λi(H)vi. (2.6)

Remarks. 1. If we “allow” infinite sums, then hHi(b)= n0 (b-1)n (Hin). 2. If 𝔤=𝔤s, then GV=xα(t).

Example. Suppose (as before) 𝔤=𝔤𝔩2 and let V=-span {(10),(01)} be the natural 𝔤-module given by matrix multiplication. Then 𝔥 has a basis 𝔥={(a00b)|a,b}= -span{(1000),(0001)}. By direct computation, xα(t)= (1t01) andh(a00b) (t)=(ta00tb) fora,b, and GV=GL2(𝔽q) (the general linear group).

Some combinatorics of the symmetric group

This section describes some of the pertinent combinatorial objects and techniques.

A pictorial version of the symmetric group Sn

Let siSn be the simple reflection that switches i and i+1, and fixes everything else. The group Sn can be presented by generators s1,s2,,sn-1 and relations si2=1, sisi+1si= si+1sisi+1, sisj=sjsi, fori-j>1. Using two rows of n vertices and strands between the top and bottom vertices, we may pictorially describe permutations in the following way. View sias
Multiplication in Sn corresponds to concatenation of diagrams, so for example,
s1 s2
Therefore, Sn is generated by s1,s2,,sn-1 with relations

Compositions, partitions, and tableaux

A composition μ=(μ1,μ2,,μr) is a sequence of positive integers. The size of μ is μ=μ1+μ2++μr, the length of μ is (μ)=r and μ= {μ1,μ2,,μr} ,whereμi= μ1+μ2++μi. (2.7) If μ=n, then μ is a composition of n and we write μn. View μ as a collection of boxes aligned to the left. If a box x is in the ith row and jth column of μ, then the content of x is c(x)=i-j. For example, if μ=(2,5,3,4)=
then μ=14, (μ)=4, μ=(2,7,10,14), and the contents of the boxes are
0 1 -1 0 1 2 3 -2 -1 0 -3 -2 -1 0
Alternatively, μ coincides with the numbers in the boxes at the end of the rows in the diagram
1 2 3 4 5 6 7 8 9 10 11 12 13 14

A partition ν=(ν1,ν2,,νr) is a composition where ν1ν2νr>0. If ν=n, then ν is a partition of n and we write νn. Let 𝒫={partitions} and𝒫n= {νn}. (2.8) Suppose ν𝒫. The conjugate partition ν=(ν1,ν2,,ν) is given by νi=Card {j|νji}. In terms of diagrams, ν is the collection of boxes obtained by flipping ν across its main diagonal. For example, ifν=

Suppose ν,μ are partitions. If νiμi for all 1i(μ), then the skew partition ν/μ is the collection of boxes obtained by removing the boxes in μ from the upper left-hand corner of the diagram λ. For example, if ν=
A horizontal strip ν/μ is a skew shape such that no column contains more than one box. Note that if μ=, then ν/μ is a partition.

A column strict tableau Q of shape ν/μ is a filling of the boxes of ν/μ by positive integers such that (a) the entries strictly increase along columns, (b) the entries weakly increase along rows. The weight of Q is the composition wt(Q)=(wt(Q)1,wt(Q)2,) given by wt(Q)i=number ofi inQ. For example, Q=
1 1 3 1 2 3 3 5
haswt(Q)= (3,1,3,0,1).

Symmetric functions

The symmetric group Sn acts on the set of variables {x1,x2,,xn} by permuting the indices. The ring of symmetric polynomials in the variables {x1,x2,,xn} is Λn(x)= {f[x1,x2,,xn]|w(f)=f,wSn}. For r0, the rth elementary symmetric polynomial is er(x:n)= 1i1<i2<<irn xi1xi2xir, where by conventioner(x:n) =0, ifr>n, and the rth power sum symmetric polynomial is pr(x:n)= x1r+x2r+ +xnr. For any partition ν𝒫, let ev(x:n) = eν1(x:n) eν2(x:n) eν(x:n) pv(x:n) = pν1(x:n) pν2(x:n) pν(x:n). The Schur polynomial corresponding to ν is sν(x:n)= det(eνi-i+j(x:n)), (2.9) for which Pieri’s rule gives sν(x:n) s(n)(x:n)= horizontal stripγ/νγ/ν=n sγ(x:n) [Mac1995, I.5.16]. (2.10)

For each t, the Hall-Littlewood symmetric function is Pν(x:n;t) = wνSnνw ( x1ν1 x2ν2 xnνn νi>νj xi-txjxi-xj ) = 1vν(t) wSnw ( x1ν1 x2ν2 xnνn i<j xi-txjxi-xj ) , where vν(t) is a constant as in [Mac1995, III.2]; and they satisfy Pν(x:n;0)= sν(x:n), P(1n) (x:n;t)= en(x:n). For t, Λn(x)=-span {eν(x:n)}= -span{sν(x:n)} =-span{Pν(x:n;t)}, (2.11) and if we extend coefficients to , then Λn(x)= -span{pν(x:n)}.

Note that for n>m there is a map Λn(x) Λm(x) f(x1,x2,,xn) f(x1,,xm,0,,0) which sends sν(x:n)sν(x:m), eν(x:n)eν(x:m), etc. Let {x1,x2,} be an infinite set of variables. The Schur function sν(x) is the infinite sequence sν(x)= ( sν(x:1), sν(x:2), sν(x:3), ) , and one can define elementary symmetric functions er(x), power sum symmetric functions pr(x), and Hall-Littlewood symmetric functions Pν(x;t), analogously. The ring of symmetric functions in the variables {x1,x2,} is Λ(x)=-span {sν(x)|ν𝒫}, and let Λ(x)= -span{sν(x)|ν𝒫}.

RSK correspondence

The classical RSK correspondence provides a combinatorial proof of the identity i,j>0 11-xiyj =νnn0 sν(x)sν(y) [Knu1970] by constructing for each 0 a bijection between the matrices bM(0) and the set of pairs (P(b),Q(b)) of column strict tableaux with the same shape. The bijection is as follows.

If P is a column strict tableau and j>0, let Pj be the column strict tableau given by the following algorithm (a) Insert j into the the first column of P by displacing the smallest number j. If all numbers are <j, then place j at the bottom of the first column. (b) Iterate this insertion by inserting the displaced entry into the next column. (c) Stop when the insertion does not displace an entry.

A two-line array (i1i2inj1j2jn) is a two-rowed array with i1i2in and jkjk+1 if ik=ik+1. If bM(0), then let b be the two-line array with bij pairs (ij).

For bM(0), suppose b= (i1i2inj1j2jn). Then the pair (P(b),Q(b)) is the final pair in the sequence (,)= (P0,Q0), (P1,Q1), (P2,Q2),, (Pn,Qn),= (P(b),Q(b)), where (Pk,Qk) is a pair of column strict tableaux with the same shape given by Pk=Pk-1jk and Qkis defined by sh(Qk)=sh(Pk) withikin the new boxsh(Qk)/sh(Qk-1). For example, b=(110002010) corresponds tob= (1122321332) and provides the sequence (,), (
, (
1 2
1 1
, (
1 2 3
1 1 2
, (
1 2 3 3
1 1 2 2
, (
1 2 3 2 3
1 1 3 2 2
so that (P(b),Q(b))= (
1 2 3 2 3
1 1 3 2 2

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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