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

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

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.

Representation theory in the G=GLn(𝔽q) case

This chapter examines the representation theory of unipotent Hecke algebras when G=GLn(𝔽q). The combinatorics associated with the representation theory of GLn(𝔽q) generalizes the tableaux combinatorics of the symmetric group, and one of the main results of this chapter is to also give a generalization of the RSK correspondence (see Section 2.3.4).

Fix a homomorphism ψ:𝔽q+*. Recall that for any composition μn, μ is a unipotent Hecke algebra (see Chapter 4). A fundamental result concerning Gelfand-Graev Hecke algebras is

([GGr1962],[Yok1968],[Ste1967]). For all n>0, (n) is commutative.

and it will follow from Theorem 5.7, via the representation theory of unipotent Hecke algebras.

The representation theory of μ

Let 𝒮 be a set. An 𝒮-partition λ=(λ(s1),λ(s2),) is a sequence of partitions indexed by the elements of 𝒮. Let 𝒫𝒮= {𝒮-partitions}. (5.1) The following discussion defines two sets Θ and Φ, so that Θ-partitions index the irreducible characters of G and Φ-partitions index the conjugacy classes of G.

Let Ln=Hom(𝔽qn*,*) be the character group of 𝔽qn*. If γLm, then let γ(r): 𝔽qmr* * x γ(x1+qr+q2r++qm(r-1)) Thus if n=mr, then we may view LmLn by identifying γLm with γ(r)Ln. Define L=n0Ln. The Frobenius maps are F: 𝔽q 𝔽q x xq and F: L L γ γq , where 𝔽q is the algebraic closure of 𝔽q.

The map {F-orbits of𝔽q*} {f𝔽q[t]|fis monic, irreducible, andf(0)0} {x,xq,xq2,,xqk-1} fx=i=1k-1(t-xqi),wherexqk=x𝔽q* is a bijection such that the size of the F-orbit of x equals the degree d(fx) of fx. Let Φ={f[t]|fis monic, irreducible andf(0)0} andΘ={F-orbits inL}. (5.2) If η is a Φ-partition and λ is a Θ-partition, then let η=fΦ d(f)η(f) andλ= φΘφ λ(φ) be the size of η and λ, respectively. Let the sets 𝒫Φ and 𝒫Θ be as in (5.1) and let 𝒫nΦ={η𝒫Φ|η=n} and𝒫nΘ= {λ𝒫Θ|λ=n}. (5.3)

(Green [Gre1955]). Let Gn=GLn(𝔽q). (a) 𝒫nΦ indexes the conjugacy classes Kη of Gn, (b) 𝒫nΘ indexes the irreducible Gn-modules Gnλ.

Suppose λ𝒫Θ. A column strict tableau P=(P(φ1),P(φ2),) of shape λ is a column strict filling of λ by positive integers. That is, P(φ) is a column strict tableau of shape λ(φ). Write sh(P)=λ. The weight of P is the composition wt(P)=(wt(P)1,wt(P)2,) given by wt(P)i= φΘφ (numbers ofiinP(φ)).

If λ𝒫Θ and μ is a composition, then let ˆμλ= {column strict tableauxP|sh(P)=λ, wt(P)=μ} (5.4) and ˆμ= {λ𝒫Θ|ˆμλis not empty}. (5.5)

The following theorem is a consequence of Theorem 2.3 and a theorem proved by Zelevinsky [Zel1981-2] (see Theorem 5.5). A proof of Zelevinsky’s theorem is in Section 5.3.

The set ˆμ indexes the irreducible μ-modules μλ and dim(μλ)= ˆμλ.

A generalization of the RSK correspondence

For a composition μn, let Nμ be as in (4.11) and Mμ as in (4.12).

The (μ,μ)-bimodule decomposition μλˆμ μλμλ impliesNμ=dim (μ)=λˆμ dim(μλ)2= λˆμ ˆμλ2. Theorem 5.4, below, gives a combinatorial proof of this identity.

Encode each matrix aMμ as a Φ-sequence (a(f1),a(f2),), fiΦ, where a(f)M(μ)(0) is given by aij(f)= highest power offdividingaij. Note that this is an entry by entry “factorization” of a such that aij=fΦ faij(f). Recall from Section 2.3.4 the classical RSK correspondence M(0) {Pairs(P,Q)of column strict tableaux of the same shape} b (P(b),Q(b)).

For aMμ, let P(a) and Q(a) be the Φ-column strict tableaux given by P(a)=(P(a(f1)),P(a(f2)),) andQ(a)= (Q(a(f1)),Q(a(f2)),) forfiΦ. Then the map Nμ Mμ { Pairs(P,Q)ofΦ-column strict tableaux of the same shape and weight ofμ } v av (P(av),Q(av)), is a bijection, where the first map is the inverse of the bijection in Theorem 4.2.

By the construction above, the map is well-defined, and since all the steps are invertible, the map is a bijection.

Example. Suppose μ=(7,5,3,2) and f,g,hΦ are such that d(f)=1, d(g)=2, and d(h)=3. Then av= ( gf2h11 h1g1 11ff2 g111 ) M
corresponds to the sequence (av(f1),av(f2),)= ( ( 0200 0000 0012 0000 ) (f) , ( 1000 0010 0000 1000 ) (g) , ( 0100 1000 0000 0000 ) (h) ) and (P(av),Q(av))= (
2 2 4 3 4
(f)
,
1 1 3
(g)
,
1 2
(h)
)
(
1 1 3 3 3
(f)
,
1 4 2
(g)
,
1 2
(h)
)
.

Zelevinsky’s decomposition of IndUG(ψμ)

This section proves the theorem

(Zelevinsky [Zel1981-2]). Let U be the subgroup of unipotent upper-triangular matrices of G=GLn(𝔽q), μn and ψμ be as in (4.8). Then IndUG(ψμ)= λˆμ Card(ˆμλ) Gλ.

Theorem 5.3 follows from this theorem and Theorem 2.3. The proof of Theorem 5.5 is in 3 steps. (1) Establish the necessary connection between symmetric functions and the representation theory of G. (2) Prove Theorem 5.5 for the case when (μ)=1. (3) Generalize (2) to arbitrary μ. The proof below uses the ideas of Zelevinsky’s proof, but explicitly uses symmetric functions to prove the results. Specifically, the following discussion through the proof of Theorem 5.7 corresponds to [Zel1981-2, Sections 9-11] and Theorem 5.5 corresponds to [Zel1981-2, Theorem 12.1].

Preliminaries to the proof (1)

Let μn and G=Gn. The group Pμ= { ( g1* g2 0g ) |giGμi =GLμi(𝔽q) } (5.6) has subgroups Lμ=Gμ1 Gμ2 Gμand Uμ= { ( Idμ1* Idμ2 0Idμ ) } , (5.7) where Idk is the k×k identity matrix. Note that Pμ=LμUμ and Pμ=NG(Uμ). The indflation map is a composition of the inflation map and the induction map, IndfLμG: R[Lμ] R[Pμ] R[G] χ InfLμPμ(χ) IndPμG(InfLμPμ(χ)), where InfLμPμ(χ): Pμ lu χ(l), forlLμand uUμ.

Suppose λ𝒫Θ and η𝒫Φ (see (5.3)). Let χλ be the irreducible character corresponding to the irreducible G-module Gλ and let κη be the characteristic function corresponding to the conjugacy class Kη (see Theorem 5.2), given by κη(g)= { 1, ifgKη, 0, otherwise, forgGη. Define R=n0R[Gn] =-span{χλ|λ𝒫Θ} =-span{κη|η𝒫Φ}. The space R has an inner product defined by χλ,χν =δλν, and multiplication χλχν= IndfL(r,s)Gr+s (χλχν), forλ𝒫rΘ, ν𝒫sΘ. (5.8)

For each φΘ, let {Y1(φ),Y2(φ),} be an infinite set of variables, and let Λ= φΘ Λ(Y(φ)), where Λ(Y(φ)) is the ring of symmetric functions in {Y1(φ),Y2(φ),} (see Section 2.3.3). For each fΦ, define an additional set of variables {X1(f),X2(f),} such that the symmetric functions in the Y variables are related to the symmetric functions in the X variables by the transform pk(Y(φ))= (-1)kφ-1 x𝔽qkφ* ξ(x)pkφd(fx) (X(fx)), (5.9) where ξφ, fxΦ is the irreducible polynomial that has x as a root, and pab(X(f))=0 if ab0. Then Λ=fΦ Λ(X(f)).

For ν𝒫, let sν(Y(φ)) be the Schur function and Pν(X(f);t) be the Hall-Littlewood symmetric function (as in Section 2.3.3). Define sλ=φΘ sλ(φ) (Y(φ))and Pη=q-n(η) fΦPη(f) (X(f);q-d(f)), (5.10) where n(η)=fΘd(f)n(η(f)) and for a composition μ, n(μ)=i=1(μ)(i-1)μi. The ring Λ=-span {sλ|λ𝒫Θ} --span{Pη|η𝒫Φ} has an inner product given by sλ,sν =δλν.

(Green [Gre1955],Macdonald [Mac1995]). The linear map ch: R Λ χλ sλ,forλ𝒫Θ κη Pη,forη𝒫Φ, is an algebra isomorphism that preserves the inner product.

A unipotent conjugacy class Kη is a conjugacy class such that η(f)= unless f=t-1. Let 𝒰=-span {κη|η(f)=, unlessf=t-1} R be the subalgebra of unipotent class functions. Note that by (5.10) and Theorem 5.6, ch(𝒰)=Λ (X(t-1)). Consider the projection π:R𝒰 which is an algebra homomorphism given by (πχλ)(g)= { χλ(g), ifgGis unipotent, 0, otherwise, λ𝒫Θ. Then π=chπch-1:ΛΛ(X(t-1)) is given by π(pk(Y(φ))) = π ( (-1)kφ-1 x𝔽qkφ* ξ(x)pkφd(fx) (X(fx)) ) (by (5.9)) = (-1)kφ-1 ξ(1)pkφ1 (X(t-1))+0 = (-1)kφ-1 pkφ (X(t-1)). (5.11)

The decomposition of IndUG(ψ(n)) (2)

The representation IndUG(ψ(n)) is the Gelfand-Graev module, and with Theorem 2.3, Theorem 5.7, below, proves that (n) is commutative.

Let U be the subgroup of unipotent upper-triangular matrices of G=GLn(𝔽q). Then ch(IndUG(ψ(n))) =λ𝒫nΘht(λ)=1 sλ,whereht(λ)= max{(λ(φ))|φΘ}.

Proof.

Let Ψ: R χλ χλ,IndUG(ψ(n)) andΨ: Λch-1 RΨ. (5.12) For any finite group H and γ,χR[H], let 1H: H C h 1 ,eH=1H hHh,and χ,γH= 1HhH χ(h)γ(h-1).

The proof is in six steps. (a) Ψ(ek(Y(1)))=δk1, where 1=1𝔽q*, (b) Ψ(χλ)=dim(e(n)Gλ) for λ𝒫Θ, (c) Ψ(fg)=Ψ(f)Ψ(g) for all f,gΛ(Y(1)), where 1=1𝔽q*. (d) Ψπ=Ψ, (e) Ψ(f(Y(φ)))=Ψ(f(Y(1))) for all fΛ(Y(φ)), (f) Ψ(sλ)=δht(λ)1.

(a) An argument similar to the argument in [Mac1995, pgs. 285-286] shows that ch-1(ek(Y(1))) =1Gk (see [HRa1999, Theorem 4.9 (a)] for details). Therefore, by Frobenius reciprocity and the orthogonality of characters, Ψ(ek(Y(1))) =1Gk,IndUG(ψ(n))= 1Uk,ψ(n)Uk= δk1.

(b) Since there exists an idempotent e such that GλGe and IndUG(ψ(n))Ge(n), the map e(n)Ge HomG(Gλ,Ge(n)) e(n)ge γg: Ge Ge(n) xe xege(n) is a vector space isomorphism (using an argument similar to the proof of [CRe1981, (3.18)]). Thus, Ψ(χλ) = χλ,IndUG(ψ(n)) = dim(HomG(Gλ,IndUG(ψ(n)))) = dim(e(n)Ge) = dim(e(n)Gλ).

(c) By (a), Ψ(er(Y(1)))Ψ(es(Y(1)))=δr1δs1. It therefore suffices to show that Ψ(er(Y(1))es(Y(1))) =δr1δs1, (sinceΛ(Y(1))= [e1(Y(1)),e2(Y(1)),]). Suppose r+s=n and let P=P(r,s). Then Ψ(er(Y(1))es(Y(1)))= Ψ(IndPGn(1P))= dim(e(n)GeP). Since TP, eP=e(1n)eP, G=vVUvU, and N=WT, e(n)GeP=e(n) Ge(1n)eP= -span{e(n)we(1n)eP|wW}. If there exists 1in such that w(i+1)=w(i)+1, then e(n)we(1n)= e(n)wxi,i+1 (t)e(1n)=e(n) xw(i),w(i)+1 (t)we(1n)= ψ(t)e(n)w e(1n). Therefore, e(n)we(1n)=0 unless w=w(n). If r>1 of s>1, then there exists 1in such that xi+1,i(t)P(r,s), so e(n)w(n)eP= e(n)w(n)xi+1,i (t)eP=e(n) xn-i,n-i+1(t) w(n)eP=ψ(t) e(n)w(n)eP=0. In particular, dim(e(n)GeP)=0. If r=s=1, then P(1,1) is upper-triangular, so e(2)w(2)eP 0 and dim(e(2)GeP)=1, giving Ψ(er(Y(1))es(Y(1)))=δr1δs1.

(d) By Frobenius reciprocity, χλ,IndUnGn(ψ(n)) = ResUnGn(χλ),ψ(n)Un = ResUnGn(π(χλ)),ψ(n) = π(χλ),IndUnGn(ψ(n)), so Ψ=Ψπ.

(e) Induct on n, using (c) and the identity (-1)n-1pn (Y(1))=nen (Y(1))- r=1n-1 (-1)r-1pr (Y(1))en-r (Y(1)), [Mac1995, I.2.11′] to obtain Ψ(pn(Y(1)))=1. Note that Ψ(pn(Y(φ))) = Ψ(π(pn(Y(φ)))) = Ψ((-1)φn-1pφn(X(t-1))) = Ψ(π(pφk(Y(1)))) = Ψ(pφk(Y(1))) = 1 = Ψ(pk(Y(1))). Since Ψ is multiplicative on Λ(Y(1)), Ψ(pν(Y(φ))) =1=Ψ(pν(Y(1))), for all partitionsν. In particular, since Ψ is linear and Λ(Y(φ))=-span{pν(Y(φ))}, Ψ(f(Y(φ)))= Ψ(f(Y(1))), for allfΛ(Y(φ)). Note that (e) also implies that Ψ is multiplicative on all of Λ.

(f) Note that Ψ(sλ)= Ψ(φΘsλ(φ)(Y(φ)))= Ψ(φΘsλ(φ)(Y(1)))= φΘΨ (sλ(φ)(Y(1))), where the last two equalities follow from (e) and (c), respectively. By definition sν(Y(1))=det(eνi-i+j(Y(1))), so Ψ(sν(Y(1)))= { 1, if(ν)=1, 0, otherwise, implies ch(IndUG(ψ(n))) =λ𝒫nΘ Ψ(sλ)sλ =λPSnΘht(λ)=1 sλ.

Decmoposition of IndUG(ψμ) (3)

Suppose λ,ν𝒫Θ. A column strict tableau P of shape λ and weight ν is a column strict filling of λ such that for each φΘ, sh(P(φ))= λ(φ)and wt(P(φ))= ν(φ). We can now prove the theorem stated at the beginning of this section:

([Zel1981-2]) Let U be the subgroup of unipotent upper-triangular matrices of G=GLn(𝔽q), μn and ψμ be as in (4.8). Then IndUG(ψμ)= λˆμ Card(ˆμλ) Gλ.

Proof.

Note that IndUPμ(ψμ) Pμeμ= Pμe[μ] e[μ], where eμ=1ULμ uULμψμ (u-1)uand e[μ]=1Uμ uUμu. (5.13) Thus IndUPμ(ψμ) InfLμPμ (IndULμLμ(ψμ)) InfLμPμ ( IndUμ1Gμ1 (ψ(μ1)) IndUμ2Gμ2 (ψ(μ2)) IndUμGμ (ψ(μ)) ) In particular, by the definition of multiplication in R (5.8), Γμ=ch (IndUG(ψμ)) =Γμ1Γμ2 Γμ,where Γμi= λ𝒫μiΘ,ht(λ)=1 sλ. Pieri’s rule (2.10) implies that for λ𝒫rΘ, ν𝒫sΘ and ht(ν)=1, sλsν= γ𝒫r+sΘ,ˆνγ/λ0 sγ,soΓμ= λ𝒫Θ Kλμsλ, where Kλμ = Card { =γ0 γ1γ2 γ= λ| ˆ(μi+1)γi+1/γi =1 } = Card{column strict tableaux of shapeλand weightμ} = ˆμλ. By Green’s Theorem (Theorem 5.6), ch is an isomorphism, so IndUG(ψμ)= ch-1(Γμ)= λˆμ ˆμλ ch-1(sλ)= λˆμ Card(ˆμλ) Gλ.

A weight space decomposition of μ-modules

Let μ=(μ1,μ2,,μ)n and let Pμ, Lμ and Uμ be as in (5.6) and (5.7). Recall that eμ=1U uUψμ (u-1)u.

For aMμ, let Ta=eμvaeμ with va as in (4.14). Then the map (μ1) (μ2) (μ) μ T(f1) T(f2) T(f) T(f1)(f2)(f), for(fi)M(μi) is an injective algebra homomorphism with image μ=eμPμeμ=eμLμeμ.

Proof.

Note that T(f1) T(f)= 1ULμ2 xi,yiUμi (i=1ψμ(xi-1yi-1)) x1v(f1)y1 xv(f)y. Since U=(LμU)(Uμ), LμUUμ1×Uμ2××Uμ, and ψμ is trivial on Uμ, T(f1)(f2)(f) = 1U2 x,yU ψμ(x-1y-1) x(v(f1)v(f2)v(f))y = 1ULμ2 xi,yiUμi ψμ(x1-1y1-1x-1y-1) e[μ]x1v(f1) y1x v(f)y e[μ], where e[μ] is as in (5.13). Since LμNG(Uμ), the idempotent e[μ] commutes with x1v(f1)y1xv(f)y and T(f1)(f2)(f) =e[μ]LU2 xi,yiUμi (i=1ψμ(xi-1yi-1)) x1v(f1)y1 xv(f)y. Consequently, the map multiplies by e[μ] and changes to , so it is an algebra homomorphism. Since the map sends basis elements to basis elements, it is also injective.

Remark. This is the GLn(𝔽q) version of Corollary 3.5.

Let μ be as in Theorem 5.8. By Theorem 5.1 each (μi) is commutative, so μ is commutative and all the irreducible μ-modules μγ are one-dimensional. Theorem 5.3 implies that ˆ(μi)= {Θ-partitionsλ|λ=μi,ht(λ)=1} indexes the irreducible (μi)-modules. Therefore, the set ˆμ= ˆ(μ1)× ˆ(μ2)×× ˆ(μ)= {γ=(γ1,γ2,,γ)|γiˆ(μi)} (5.14) indexes the irreducible μ-modules. Identify γˆμ with the map γ:μ such that yv=γ(y)v, for allyμ, vμγ.

For γˆμ, the γ-weight space Vγ of an μ-module V is Vγ= { vV|yv=γ (y)v, for allyμ } . Then Vγˆμ Vγ.

Let λ𝒫Θ and γˆμ. A column strict tableau P of shape λ and weight γ is column strict filling of λ such that for each φΘ, sh(P(φ))= λ(φ)and wt(P(φ))= ( γ1(φ), γ2(φ),, γ(φ) ) , where γi(φ) is the number of boxes in the partition γi(φ) (which has length 1). Let ˆγλ= { Pμλ|sh (P)=λ, wt(P)=γ } .

For example, suppose λ= (
(φ1)
,
(φ2)
,
(φ3)
)
and γ= (
(φ1)
,
(φ2)
,
(φ3)
)
(
(φ1)
,
(φ3)
)
(
(φ1)
,
(φ2)
)
(
(φ2)
)
.
Then ˆγλ= { (
1 1 3 2 2
(φ1)
,
1 3 4
(φ2)
,
1 2
(φ3)
)
,
(
1 1 3 2 2
(φ1)
,
1 4 3
(φ2)
,
1 2
(φ3)
)
(
1 1 2 2 3
(φ1)
,
1 3 4
(φ2)
,
1 2
(φ3)
)
,
(
1 1 2 2 3
(φ1)
,
1 4 3
(φ2)
,
1 2
(φ3)
)
}
.

Let μλ be an irreducible μ-module and γˆμ. Then dim(μλ)γ= Card(ˆγλ).

Proof.

By Theorem 2.3 and Proposition 2.2, dim((μλ)γ)= Resμμ(μλ),μγ= ResPμG(Gλ),Pμγ= Gλ,IndPμG(Pμγ), where Pμγ=InfLμPμ(Lμγ). Therefore, dim((μλ)γ)= cγλ,where sγ1sγ2 sγ=λ𝒫Θ cγλsλ. Pieri’s rule (2.10) implies cγλ=ˆγλ.

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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