Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type

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

Last update: 8 June 2014

Notes and References

This is a typed copy of Peter Norbert Hoefsmit's thesis Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type published in August, 1974.

Chapter 2.Representations of the Generic Ring Corresponding to a Coxeter System of Classical Type

Partitions and Tableaux

Let B be a subgroup of a finite group G and let k be a field of characteristic zero. Set e=|B|-1bBb in the group algebra kG. Then e affords the 1-representation 1B of B and the left kG-module kGe affords the induced representation 1BG.

Definition (2.1.1) The Hecke algebra Hk(G,B) is the subalgebra of kG given by e(kG)e.

The Hecke algebra acts on kGe by right multiplication and the action defines an isomorphism between Hk(G,B) and the endomorphism algebra EndkG(kGe). The double coset sums xBgBx, gG, form a basis for Hk(G,B) (see [Ste1967], Lemma 84).

In this thesis we will be concerned with finite groups G with BN-pairs of subgroups (B,N) satisfying the axioms of [Tit1969]. Then H=BN is a normal subgroup of N and the Weyl group W=N/H has a presentation with a set of distinguished involutionary generators R and defining relations (2.1.2) r2=1,rR (rs)nrs= (sr)nrs, r,sR,rs where nrs is the order of rs in W and (xy)m denotes a product of alternating x's and y's with m factors. The pair (W,R) is called a Coxeter system. The group G is said to be of type (W,R).

If w1W, we denote by l(w) the least length l of all expressions (2.1.3) w=r1rl, r1,,rlR. (2.1.3) is called a reduced expression for W if l=l(w).

There is a bijection between the double cosets B\G/B and the elements wW resulting in the Bruhat decomposition G=wWBwB. The structure of the Hecke algebra Hk(G,B) of a finite group with a BN-pair with respect to a Borel subgroup B was shown in [Iwa1964] and [Mat1969-2] to be as follows.

Theorem (2.1.4) Hk(G,B) has k-basis {Sw:aW} where Sw=|B|-1 xBwBx with S1 the identity element. Multiplication is determined by the formula SwSr = Swr,rR, l(wr)>l(w), SwSr = qrSwr+ (qr-1)Sw, rR,l(wr)<l(w) where the {qr,rR} are the index parameters (2.1.5) qr=|B:(BrBr)|. For any reduced expression w=r1rl for W in R, w1 Sw=Sr1 Srl. Thus Hk(G,B) is generated by {Sr,rR} and has defining relations (2.1.6) Sr2=qrS1 +(qr-1)Sr, rR (SrSs)nrs= (SsSr)nrs, where nrs is as in (2.1.2).

Let (W,R) be a Coxeter system and let {μr,rR} be indeterminates over k, chosen such that μr=μs if and only if r and s are conjugate in W. Let D be the polynomial ring D=k[μr:rR]. Then there exists an associative D-algebra 𝒜 with identity, free basis {aw,wW} over D and multiplication determined by the formulas (2.1.7) awar = awr,rR, l(wr)>l(w), awar = μrawr+ (μr-1)aw, rR,l(wr)<l(w), (see [Bou1968], p. 55). The D-algebra 𝒜 is called the generic ring corresponding to the Coxeter system (W,R). Analogous to Theorem (2.1.4) the generic ring has a presentation with generators {ar,rR} and relations (2.1.8) ar2 = μra1+ (μr-1)ar, rR (aras)nrs = (asar)nrs ,r,sR,rs with nrs is as in (2.1.2).

The Hecke algebra Hk(G,B) can be compared with the group algebra kW as follows. Let L be any field of characteristic zero and ϕ:DL a homomorphism. Consider L as a D-module by setting d·λ=ϕ(d)λ ,dD,λL. Then the specialized algebra (2.1.9) 𝒜ϕ,L=L𝒜 is an algebra over L with basis {awϕ=1aw}, generators {arϕ,rR} and defining relations obtained from (2.1.8) by applying ϕ. Thus if ϕ:Dk is defined by ϕ(μr)=qr, rR, qr the index parameters (2.1.5), then (2.1.10) 𝒜ϕ,k=Hk (G,B) while if ϕ0:Dk is defined by ϕ0(μr)=1, for all rR, then (2.1.11) 𝒜ϕ0,k=kW.

We say the Coxeter system (W,R) is of classical type if W is of type An, Bn, n2, or Dn, n4. In this chapter we will determine the irreducible representations of the generic ring corresponding to a Coxeter system of classical type and by means of the appropriate specialized algebras the irreducible representations of the Hecke algebras Hk(G,B) of groups with BN-pair of classical type.

The Representations of 𝒜K(Bn)

If a Coxeter system (W,R) is of type Bn, n2, W(Bn) is isomorphic to the hyperoctahedral group, the group of signed permutations on n letters (see 2). Thus W(Bn) has a presentation with generators R={w1,,wn} where wi=(i-1,i), i=2,,n, and w1=-(1), the first sign change and relations wi2 = 1, w1w2w1w2 = w2w1w2w1, wiwi+1wi = wi+1wiwi+1, i=2,,n-1; wiwj = wjwi,|i-j|>1 (see [Car1972]). Furthermore the set of generators R is partitioned into 2 sets under conjugation; namely, wi is conjugate to wj for i,j2 while the negative one-cycle w1 is not conjugate to any wj, j2.

For the Coxeter system (W(Bn),R) taken as above, we take the generic ring 𝒜(Bn) to be defined over the polynomial ring D=[x,y], x,y indeterminates over . It has a presentation with generators awi=ai, aiR, and relations

(B1) a12=y1+(y-1)a1,
(B2) ai2=x1+(x-1)ai,i=2,,n;
(B3) a1a2a1a2=a2a1a2a1,
(B4) aiaj=ajai,|i-j|>1.
We depart from the notations of 1 strictly for notational convenience, i.e., we switch from (μ1,μ2) to (x,y) to avoid carrying around subscripts.

We now construct for each double partition (μ)=(α,β) of n, n2, a k-representation of 𝒜K(Bn)=K𝒜(Bn), K=(x,y). The method involves the construction of fμ×fμ matrices over k for each of the generators ai of 𝒜K(Bn) in a manner analogous to the construction of the matrices of the transpositions (i-1,i) for the outer tensor product representation [α]·[β] of Sn.

For any integer k, let Δ(k,y)=xk y+1 Denote by M(k,y) the 2×2 matrix (2.2.1) M(k,y)=1Δ(k,y) ( (x-1) Δ(k+1,y) xΔ(k-1,y) xky(x-1) ) . Then traceM(k,y)=(x-1), detM(k,y)=-x, so the characteristic polynomial of M(k,y) gives (2.2.2) M(k,y)2=xI+ (x-1)M(k,y), I the 2×2 identity matrix.

For k1, let Δ(k,-1)= i=0k-1 xi. Denote by M(k,-1), k2, the 2×2 matrix (2.2.3) M(k,-1)= 1Δ(k,-1) ( -1 Δ(k+1,-1) xΔ(k-1,-1) xk ) . As M(k,-1) is obtained from M(k,y) by setting y=-1, (2.2.1) shows (2.2.4) M(k,-1)2=xI +(x-1)M(k,-1) Denote by D(z,w) the 2×2 diagonal matrix D(z,w)= ( z0 0w ) . Then (2.2.5) D(z,-1)2=zI +(z-1)D(z,-1). In what follows, we employ the definitions and notations of (1) in regards to double partitions, standard tableaux, and axial distance.

Definition (2.2.6) Let (μ)=(α,β) be a double partition of n and let T1μ,,Tfμ, f=fμ be the ordering of the standard tableaux of shape (μ) according to the last letter sequence. Construct f×f matrices Mμ(i), i=1,,n, over K=(x,y) as follows:

(1) Construct Mμ(1) by placing
(i) y in the p,p-th entry if the letter 1 appears in Tpα of Tpμ=(Tpα,Tpβ),
(ii) -1 in the p,p-th entry if the letter 1 appears in Tpβ of Tpμ=(Tpα,Tpβ),
(iii) zeros elsewhere.
(2) Construct Mμ(i), i=2,,n, by placing
(i) x in the p,p-th entry if the letters i-1 and i appear in the same row of Tpα or Tpμ=(Tpα,Tpβ),
(ii) -1 in the p,p-th entry if the letters i-1 and i appear in the same column of Tpα or Tpβ of Tpμ,
(iii) the matrix M(k,-1) in the p,p-th, p,q-th,, q,p-th and q,q-th entries corresponding to the tableaux Tpμ and Tqμ where
(a) p<q, (i-1,i)Tpμ=Tqμ and the letters i-1 and i appear either both in Tpα or Tpβ of Tpμ,
(b) k is the axial distance from i to i-1 in Tpμ,
(iv) the matrix M(k,y) in the p,p-th, p,q-th, q,p-th and q,q-th entries corresponding to the tableaux Tpμ and Tqμ where
(a) p<q, (i-1,i)Tpμ=Tqμ and the letters i-1 and i appear in different tableaux of Tpμ,
(b) k is the axial distance from i to i-1 in Tpμ,
(v) zeros elsewhere.

Let Vμ denote the free -module generated by t1,,tf, f=fμ corresponding to the standard tableaux T1μ,,Tfμ of shape (μ) ordered according to the last letter sequence. For any field L of characteristic zero set VμL=VμL. The corresponding basis elements ti1 of VμL will be denoted simply by ti. Set K=(x,y). Define linear operators Ziμ, i=1,,n, on VμK such that the matrix of Ziμ with respect to the basis {t1,,tf} of VμK is given by Mμ(i).

Theorem (2.2.7) Let K=(x,y) and let 𝒜K(Bn) denote the generic ring of the Coxeter system (W(Bn),R) as before. Let (μ) be a double partition of n, n2. Then the K-linear map πμ :𝒜K(Bn) End(VμK) defined by πμ(ai)=Ziμ is a representation of 𝒜K(Bn).

Proof.

Theorem (2.2.14) Let k be as before. The representations πμ of 𝒜K(Bn) are irreducible, pairwise inequivalent and are, up to isomorphism, a complete set of irreducible, inequivalent representations of 𝒜K(Bn). In particular k is a splitting field for 𝒜K(Bn).

Proof.

It is clear that the above representations of the generic ring yield representations of a wide variety of specialized algebras of 𝒜K(Bn). Specifically, set P(Bn)=x i=0n-1 (xi+y) (xiy+1) (1++xi)D =[x,y].

Corollary (2.2.15) Let L be any field of characteristic zero, ϕ:DL a homomorphism such that ϕ(P(Bn))0. Let (μ) be a double partition of n and let Ziϕμ denote the linear operator on VμL obtained by the substitution xϕ(x), yϕ(y) in the entry of Mμ(i). Then Ziϕμ is well defined and the L-linear map πϕ,Lμ: 𝒜ϕ,L(Bn) End(VμL) defined by πϕ,Lμ(ai)=Ziϕμ is a representation of 𝒜ϕ,L(Bn). The representations {πϕ,Lμ} are a complete set of irreducible inequivalent representations of 𝒜ϕ,L(Bn).

Proof.

Let A be a separable algebra over a field L and let L be an algebraic closure of L. Define the numerical invariants of A to be the set of integers {ni} such that AL is isomorphic to a direct sum of total matrix algebras AL=i Mni(L). Thus for ϕ defined as in Corollary (2.2.15) the algebras 𝒜K(Bn) and 𝒜ϕ,L have the same numerical invariants. In particular Corollary (2.2.15) gives the well known result (see [BCu1972]) that for G a finite group with BN-pair with Coxeter system (W,R) of type Bn, H(G,B) W. Indeed in ([BCu1972]) this is shown to be the case for all Coxeter system with the possible exception of (W,R) of type E7.

Finally we remark that, for the specialization ϕ0:D defined by ϕ0(x)=ϕ0(y)=1, the representations {πϕ0,μ} are the irreducible representations of W(Bn) given by Theorem (1.2.3).

The Representations of 𝒜K(An) and 𝒜K(Dn)

We now obtain the representations of the generic ring of a Coxeter system of type An and Dn.

If (W,R) is a Coxeter system of type An-1, W(An-1) is isomorphic to the symmetric group Sn and we take the set R to be {w2,,wn} where wi=(i-1,i), i=2,,n. We take the generic ring 𝒜(An-1) to be defined over the polynomial ring D=[x]. It has a presentation with generators awi=ai, i=2,,n, and relations (B2, B4, B5).

Set K=(x). The representations of 𝒜K(An-1)=𝒜(An-1)DK are readily obtained from the results of the previous section. As the matrices M(k,-1) are defined in (x), (2.2.6) shows the matrices M(α,(0))(i), i=2,,n, are defined in (x) and Zi(α,(0)) can be regarded as a linear operator on V(α,(0))K. Thus

Theorem (2.3.1) Let α be a partition of n, n2 and K=(x). The k-linear map πα:𝒜K (An-1) End(V(α,(0))K) defined by πα(ai)=Zi(α,(0)), i=2,,n, is a representation of 𝒜K(An-1). The representations {πα}, are a complete set of irreducible, inequivalent representations of 𝒜K(An-1).

Proof.

The representations of the specialized algebra are handled entirely analogous to Corollary (2.2.15). Set P(An)=x i=1n (1++xi). Then from the above and Corollary (2.2.15) we have

Corollary (2.3.2) Let L be any field of characteristic zero, ϕ:D=[X]L a homomorphism such that ϕ(P(An))0. Then for (α) a partition of n2, the linear operators Ziϕ(α,(0)), i=2,,n, are well defined and the L-linear maps πϕ,Lα: 𝒜ϕ,L (An-1) End(V(α,(0))L) defined by πϕ,Lα(ai)=Ziϕ(α,(0)) is a representation of 𝒜ϕ,L(An-1). The {πϕ,Lα} are a complete set of irreducible, inequivalent representations of 𝒜ϕ,L(An-1).

Thus for ϕ as above the algebras 𝒜K(An) and 𝒜ϕ,L(An) have the same numerical invariants. We remark that for the specialization x1 the definitions of the matrices M(k,-1) shows the semi-normal matrix representation of Sn is obtained (see Theorem (1.2.1)).

If (W,R) is a Coxeter system of type Dn, n4, W(Dn) can be regarded as a subgroup of index 2 in W(Bn); W(Dn) acting on an orthonormal basis of n by means of permutations and even sign changes. A set of distinguished generators for W(Dn) can be obtained from the set {w1,,wn} of W(Bn) given in section (1) by setting w1=w1w2w1 and taking the set R to be {w1,w2,,wn}. (see [Car1972]).

Let ϕ:[x,y][x] be defined by ϕ(y)=1. Then the specialized ring 𝒜ϕ,[x](Bn) has basis {awϕ,wW(Bn)} with relations obtained by applying ϕ to (B1 - B5). In particular (a1ϕ)2=1. Set a1ϕ= aw1w2w1ϕ. As w1w2w1 is reduced in (W(Bn),R) we have aw1w2w1ϕ=a1ϕa2ϕa1ϕ by (2.1.8). Applying ϕ to (B1 - B5) it is readily seen that

(B'1) a1ϕ2=x1+(x-1)a1ϕ,
(B'2) a1ϕa3ϕa1ϕ=a3ϕa1ϕa3ϕ,
(B'3) a1ϕajϕ=ajϕa1ϕ,j1,3.
As any reduced expression of w1W(Dn) in the generators {w1,w2,,wn} is a reduced expression for W in the generators {w1,w2,,wn} of W(Bn), the relations (B'1 - B'3) show the subring of 𝒜ϕ,[x](Bn) generated by {a1ϕ,a2ϕ,,anϕ} has free basis {awϕ,wW(Dn)}. As all the generators {w1,w2,,wn} are conjugate in W(Dn), the subring generated by {a1ϕ,a2ϕ,,anϕ} is isomorphic to the generic ring of a Coxeter system of type Dn, n4. Denote this subring by 𝒜(Dn).

Thus the representations of 𝒜ϕ,L(Bn) given by Corollary (2.2.15) provide us with representations of 𝒜K(Dn). Young ([You1929]) showed the restrictions of the representations of W(Bn) to W(Dn) corresponding to a double partition (α,β) of n remain irreducible if (α)(β) and decomposes into two irreducible components when (α)=(β). We show that this holds true in a generic sense.

Recall that a standard tableau T for the double partition (α,β) of n is an ordered pair T=(Tα,Tβ). Then the tableau T*=(Tβ,Tα) is a standard tableau of shape (β,α), called the conjugate tableau of T. Moreover the map TT* is a bijection from the standard tableaux of shape (α,β) to the standard tableaux of shape (β,α). Take (α)(β). If T1,Tp,,Tq,Tf, f=fα,β, is the arrangement of the standard tableaux of shape (α,β) according to the last letter sequence, order the tableaux of shape (β,α) according to the scheme; Tq* precedes Tp* if Tp precedes Tq in the last letter sequence. Call this the conjugate ordering of the tableaux of shape (β,α).

Let In* denote the n×n matrix In*= ( 0 0 1 0 1 0 1 0 0 ) .

Lemma (2.3.3) Let Mϕα,β(a) denote the matrix of πϕα,β(a) with respect to the basis {ti} of Vα,βK ordered according to the last letter sequence, a𝒜K(Dn). Then If*Mϕα,β(a)If*, f=fα,β, is the matrix of πϕβ,α(a) with respect to the conjugate ordering of the basis {ti} of Vβ,αK. Thus the restrictions of the representations πϕα,β and πϕβ,α to 𝒜K(Dn) are equivalent.

Proof.

We define the conjugate ordering of the standard tableaux T=(T1α,T2α) of shape (α,α) as follows. Set 𝒯i= {T=(T1α,T2α):nappears inTiα} ,i=1,2. All standard tableaux belonging to 𝒯2 precede those belonging to 𝒯1 in the arrangement according to the last letter sequence. Rearrange the last 12fα,α tableaux in the last letter sequence, i.e. those in 𝒯1, as follows; for T1, T2 in 𝒯1, Ta precedes T2 if T2* precedes T1* in the last letter sequence arrangement of the tableaux in 𝒯2.

Lemma (2.3.5) Let Mπα,α(a) denote the matrix of πϕα,α(a) on Vα,αK with respect to the conjugate ordering of the basis {ti} of Vα,αK, a𝒜K(Dn). Set Rf= ( -I12f I12f* I12f* I12f ) ,f=fα,α Then (2.3.6) RfMϕα,α Rf-1= ( M1(a)0 0M2(a) ) .

Proof.

Let Vα,αK= 1Vα,αK= 2Vα,αK= where for basis elements tp corresponding to tableaux Tp𝒯2, 1Vα,αK has basis {tp+tp*} and 2Vα,αK has basis {tp-tp*}. By Lemma (2.3.5) the k-linear maps iπϕα,α: 𝒜K(Dn)End (Vα,αK) where the matrix of iπϕα,α(aw) with respect to the above basis is Mi(aw), wW(Dn), are representations of 𝒜K(Dn).

Theorem (2.3.9) For double partitions (α,β), (α,β), |α|<|β|, and (α,α) of n4, the representations πϕα,β and iπϕα,α, i=1,2, are a complete set of irreducible, inequivalent representations of 𝒜K(Dn).

Proof.

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