Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

DOI 10.1007/s10801-007-0058-3

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

John Enyang

Received: 26 April 2005 / Accepted: 8 January 2007 / Published online: 7 April 2007

© Springer Science+Business Media, LLC 2007

Abstract A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra Bn(q, r)by lifting bases for cell modules of Bn−1(q, r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke alge- bra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parametersq andr, for B–M–W algebras to be semisimple. The aforementioned constructions pro- vide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori–Hecke algebra of the symmetric group.

Keywords Birman–Murakami–Wenzl algebra·Brauer algebra·Specht module· Cellular algebra·Jucys–Murphy operators

1 Introduction

Using a recursive procedure which lifts bases ofBi−1(q, r)to bases forBi(q, r), for i=1,2, . . . , n, we obtain new cellular bases (in the sense of [5]) for the B–M–W algebraBn(q, r), indexed by paths in an appropriate Bratteli diagram, whereby 1. each cell module forB_n(q, r)admits a filtration by cell modules forB_n−1(q, r),

and

2. certain commuting elements inB_n(q, r), which generalise the Jucys–Murphy el- ements in the Iwahori–Hecke algebra of the symmetric group, act triangularly on each cell module for the algebraB_n(q, r).

Research supported by Japan Society for Promotion of Science.

J. Enyang

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

The triangular action of the generalised Jucys–Murphy elements, combined with the machinery of cellular algebras from [5], allows us to obtain explicit criteria, in terms of defining parameters, for any given B–M–W algebra to be semisimple. The afore- mentioned provide generalisations of classical results from the representation theory of the Iwahori–Hecke algebra of the symmetric group to the algebras under investi- gation here.

The contents of this article are presented as follows.

1. Definitions concerning partitions and tableaux, along with standard facts from the representation theory of the Iwahori–Hecke algebra of the symmetric group are stated in Sect.2.

2. In Sect.3, we define a generic version of the B–M–W algebras and restate in a more transparent notation the main results of [4] on cellular bases of the same algebras.

3. In Sect. 4, we state for reference some consequences following from the state- ments in Sect.3and the theory of cellular algebras given in [5].

4. In Sect.5, an explicit description of the behaviour of the cell modules for generic B–M–W algebras under restriction is obtained.

5. In Sect.6, the results of Sect.5are used to construct new bases for B–M–W alge- bras, indexed by pairs of paths in the Bratteli diagram associated with B–M–W al- gebras and generalising Murphy’s construction [9] of bases for the Iwahori–Hecke algebras of the symmetric group. A demonstration of the iterative procedure is given in detail in Examples6.2and6.3.

6. Certain results of R. Dipper and G. James on the Jucys–Murphy operators of the Iwahori-Hecke algebra of the symmetric group are extended to generic B–M–W algebras in Theorem7.8.

7. Theorems8.2and8.5use the above mentioned results to give sufficient criteria for the B–M–W algebras over a field to be semisimple.

8. Theorem10.7shows that the Jucys–Murphy elements act triangularly on each cell module of the Brauer algebra, while the semisimplicity criterion of Theorem11.1 is a weak version of a result of H. Rui [11].

9. Some conjectures on the semisimplicity of the Brauer algebras are given in Sect.12.

The author is indebted to B. Srinivasan for guidance, to A. Ram for remarks on a previous version of this paper, and to I. Terada for discussions during the period this work was undertaken. The author is grateful to T. Shoji and H. Miyachi for comments and thanks the referees for numerous suggestions and corrections.

2 Preliminaries

2.1 Combinatorics and tableaux

Throughout, n will denote a positive integer andS_n will be the symmetric group acting on{1, . . . , n}on the right. Forian integer, 1≤i < n, lets_i denote the transpo- sition(i, i+1). ThenS_nis generated as a Coxeter group bys₁, s₂, . . . , s_n−1, which

satisfy the defining relations

s_i²=1 for 1≤i < n;

sisi+1si=si+1sisi+1 for 1≤i < n−1;

s_is_j=s_js_i for 2≤ |i−j|.

An expressionw=s_i1s_i2· · ·s_ik in whichkis minimal is called a reduced expression forw, and(w)=kis the length ofw.

Let f be an integer, 0≤f ≤ [n/2]. If n−2f >0, a partition ofn−2f is a non–increasing sequenceλ=(λ₁, . . . , λ_k)of integers,λ_i ≥0, such that_k

i=1λ_i = n−2f; otherwise, if n−2f =0, writeλ=∅ for the empty partition. The fact that λis a partition of n−2f will be denoted byλn−2f. We will also write

|λ| =

i≥1λi. The integers{λi :fori≥1}are the parts ofλ. Ifλis a partition of n−2f, the Young diagram ofλis the set

[λ] = {(i, j ):λ_i ≥j≥1 andi≥1} ⊆N×N.

The elements of[λ]are the nodes ofλand more generally a node is a pair(i, j )∈ N×N. The diagram [λ] is traditionally represented as an array of boxes withλ_i boxes on thei–th row. For example, ifλ=(3,2), then[λ] = . Let[λ]be the diagram of a partition. A node(i, j ) is an addable node of[λ] if (i, j )∈ [λ] and [μ] = [λ] ∪ {(i, j )}is the diagram of a partition; in this case(i, j )is also referred to as a removable node of[μ].

For our purposes, a dominance order on partitions is defined as follows: ifλand μare partitions, thenλμif either

1. |μ|>|λ|or 2. |μ| = |λ|and_k

i=1λ_i≥_k

i=1μ_i for allk >0.

We will writeλμ to mean thatλμandλ=μ. Although the definition of the dominance order on partitions employed here differs from the conventional defini- tion [7] of the dominance order on partitions, when restricted to the partitions of the odd integers{1,3, . . . , n}or to partitions of the even integers{0,2, . . . , n}, depend- ing asn is odd or even, the orderas defined above is compatible with a cellular structure of the Birman–Murakami–Wenzl and Brauer algebras, as shown in [4], [5]

and [13].

Letf be an integer, 0≤f ≤ [n/2], andλbe a partition ofn−2f. Aλ–tableau labeled by{2f+1,2f+2, . . . , n}is a bijectiontfrom the nodes of the diagram[λ] to the integers{2f +1,2f+2, . . . , n}. A givenλ–tableaut: [λ] → {2f +1,2f + 2, . . . , n}can be visualised by labeling the nodes of the diagram[λ]with the integers 2f+1,2f+2, . . . , n. For example, ifn=10,f =2 andλ=(3,2,1),

t= (2.1)

represents aλ–tableau. Aλ–tableautlabeled by{2f+1,2f +2, . . . , n}is said to be standard if

t(i₁, j₁)≥t(i₂, j₂), wheneveri₁≥i₂andj₁≥j₂.

Ifλis a partition ofn−2f, write Stdn(λ)for the set of standardλ–tableaux labeled by the integers{2f+1,2f +2, . . . , n}. We lett^λ denote the element of Stdn(λ)in which 2f +1,2f+2, . . . , nare entered in increasing order from left to right along the rows of[λ]. Thus in the above example wheren=10,f =2 andλ=(3,2,1),

t^λ= . (2.2)

The tableaut^λis referred to as the superstandard tableau in Stdn(λ). Ift∈Stdn(λ), we will write λ=Shape(t) and, abiding by the convention used in the literature, Std(λ) will be used to denote the set of standard tableauxt: [λ] → {1,2, . . . ,|λ|};

we will refer to elements of Std(λ)simply as standardλ–tableaux. Ifs∈Stdn(λ), we will writesˆfor the tableau in Std(λ)which is obtained by relabelling the nodes ofs by the mapi→i−2f.

If t∈Stdn(λ) and i is an integer 2f < i≤n, define t|i to be the tableau ob- tained by deleting each entrykoftwithk > i(compare Example5.1below). The set Stdn(λ)admits an orderwhereinstif Shape(s|i)Shape(t|i)for each integer iwith 2f < i≤n. We adopt the usual convention of writingstto mean thatst ands=t.

The subgroupSn−2f = si:2f < i < n ⊂Snacts on the set ofλ–tableaux on the right in the usual manner, by permuting the integer labels of the nodes of[λ]. For example,

(6,8)(7,10,9)= . (2.3)

Ifλ is a partition ofn−2f, then for our purposes the Young subgroupSλ is de- fined to be the row stabiliser oft^λinSn−2f. For instance, whenn=10,f =2 and λ=(3,2,1), as in (2.2) above, thenSλ= s5, s6, s8. To eachλ–tableaut, associate a unique permutationd(t)∈Sn−2f by the conditiont=t^λd(t). If we refer to the tableautin (2.1) above for instance, thend(t)=(6,8)(7,10,9)by (2.3).

2.2 The Iwahori–Hecke algebra of the symmetric group

For the purposes of this section, letRdenote an integral domain andqbe a unit inR. The Iwahori–Hecke algebra (overR) of the symmetric group is the unital associative R–algebraHn(q²)with generatorsX1, X2, . . . , Xn−1,which satisfy the defining re- lations

(X_i−q)(X_i+q⁻¹)=0 for 1≤i < n;

X_iX_i+1X_i=X_i+1X_iX_i+1 for 1≤i < n−1;

X_iX_j=X_jX_i for 2≤ |i−j|.

Ifw∈Snandsi₁si₂· · ·si_k is a reduced expression forw, then X_w=X_i1X_i2· · ·X_ik

is a well defined element of Hn(q²)and the set {X_w : w∈S_n}freely generates Hn(q²)as anR–module (Theorems 1.8 and 1.13 of [8]).

Below we state for later reference standard facts from the representation theory of the Iwahori–Hecke algebra of the symmetric group, of which details can be found in [8] or [9]. Ifμis a partition ofn, define the element

cμ=

w∈Sμ

q^l(w)Xw.

In this section, let∗denote the algebra anti–involution ofHn(q²)mappingX_w→ X_w−1. If λis a partition of n,Hˇ_n^λ is defined to be the two–sided ideal inHn(q²) generated by

cuv=X_d(u)^∗ cμXd(v):u,v∈Std(μ), whereμλ . The next statement is due to E. Murphy in [9].

Theorem 2.1 The Iwahori–Hecke algebraHn(q²)is free as anR–module with basis M=

c_uv=X^∗_d(u)c_λX_d(v)

foru,v∈Std(λ)and λa partition ofn

.

Moreover, the following statements hold.

1. TheR–linear anti–involution∗satisfies∗ :c_st→c_tsfor alls,t∈Std(λ).

2. Suppose that h∈Hn(q²), and thatsis a standardλ–tableau. Then there exist au∈R, foru∈Std(λ), such that for allv∈Std(λ),

cvsh≡

u∈Std(λ)

aucvu modHˇ_n^λ. (2.4)

The basis M is cellular in the sense of [5]. If λis a partition of n, the cell (or Specht) moduleC^λforHn(q²)is theR–module freely generated by

{c_s=c_λX_d(s)+ ˇH_n^λ :s∈Std(λ)}, (2.5) and given the rightHn(q²)–action

csh=

u∈Std(λ)

aucu, forh∈Hn(q²),

where the coefficientsau∈R, foru∈Std(λ), are determined by the expression (2.4).

The basis (2.5) is referred to as the Murphy basis forC^λandMis the Murphy basis forHn(q²).

Remark 2.1 TheHn(q²)–moduleC^λis the contragradient dual of the Specht module defined in [2].

Let λ and μ be partitions of n. A λ–tableau of type μ is a map T: [λ] → {1,2, . . . , d}such that μ_i= |{y∈ [λ] :^T(y)=i}|fori≥1. Aλ–tableauTof type μis said to be semistandard if (i) the entries in each row ofTare non–decreasing, and (ii) the entries in each column ofTare strictly increasing. Ifμis a partition, the semistandard tableau^Tμis defined to be the tableau of typeμwith^Tμ(i, j )=ifor (i, j )∈ [μ].

Example 2.1 Let μ =(3,2,1). Then the semistandard tableaux of type μ are

T^μ= , , , , , , , and

, as in Example 4.1 of [8]. All the semistandard tableaux of typeμare obtainable fromT^μby “moving nodes up” inT^μ.

Ifλandμare partitions ofn, the set of semistandardλ–tableaux of typeμwill be denoted byT0(λ, μ). Further, given aλ–tableautand a partitionμofn, thenμ(t)is defined to be theλ–tableau of typeμobtained fromtby replacing each entryiint withkifiappears in thek–th row of the superstandard tableaut^μ∈Std(μ).

Example 2.2 Letn=7, andμ=(3,2,1,1), so thatt^μ= . Ifν=(4,3)and

t= , thenμ(t)= .

Letμandνbe partitions ofn. IfSis a semistandardν–tableau of typeμ, andtis a standardν–tableau, define inHn(q²)the element

cSt=

s∈Std(ν) μ(s)=^S

q^(d(s))cst. (2.6)

Given a partitionμofn, letM^μbe the rightHn(q²)–module generated byc_μ. The next statement is a special instance of a theorem of E. Murphy (Theorem 4.9 of [8]).

Theorem 2.2 Letμbe a partition ofn. Then the collection

{cSt:^S∈T0(ν, μ),t∈Std(ν), forνa partition ofn} freely generatesM^μas anR–module.

Ifμandλare partitions ofn−1 andnrespectively, for the purposes of the present Sect.2.2, we writeμ→λto mean that the diagram[λ]is obtained by adding a node to the diagram[μ], as exemplified by the truncated Bratteli diagram associated with

Hn(q²)displayed in (2.7) below (Sect. 4 of [6]).

(2.7) Ifλis a partition ofnthen, as in [6], define a path of shapeλin the Bratteli diagram associated withHn(q²)to be a sequence of partitions

λ⁽⁰⁾, λ⁽¹⁾, . . . , λ⁽ⁿ⁾

satisfying the conditions thatλ⁽⁰⁾=∅is the empty partition,λ⁽ⁿ⁾=λ, andλ⁽ⁱ⁻¹⁾→ λ⁽ⁱ⁾, for 1≤i≤n. As observed in Sect. 4 of [6], there is a natural correspondence between the paths in the Bratteli diagram associated withHn(q²)and the elements of Std(λ)wherebyt→(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ⁽ⁿ⁾)andλ⁽ⁱ⁾=Shape(t|i)for 1≤i≤n.

Example 2.3 Let n=6 and λ=(3,2,1). Then the identification of standard λ–

tableau with paths of shapeλin the Bratteli diagram associated withHn(q²)maps

t= →

, , , , ,

.

Taking advantage of the bijection between the standardλ–tableaux and the paths of shapeλin the Bratteli diagram ofHn(q²), we will have occasion to write

t= λ⁽⁰⁾, λ⁽¹⁾, . . . , λ⁽ⁿ⁾

,

explicitly identifying each standardλ–tableautwith a path of shapeλin the Bratteli diagram.

For each integeriwith 1≤i≤n, considerHi(q²)as the subalgebra ofHn(q²) generated by the elementsX₁, X₂, . . . , X_i−1, thereby obtaining the tower of algebras R=H1(q²)⊆H2(q²)⊆ · · · ⊆Hn(q²). (2.8) Given a rightHn(q²)-moduleV, write Res(V )for the restriction ofV toHn−1(q²) by the identifications (2.8). Lemma2.3below, which is a consequence of Theorem 7.2 of [9], shows that the Bratteli diagram associated withHn(q²)describes the behav- iour of the cell modules forHn(q²)under restriction toHn−1(q²).

Lemma 2.3 Letλbe a partition ofn. For each partitionμofn−1 withμ→λ, let A^μdenote theR–submodule ofC^λfreely generated by

{c_v:v∈Std(λ)and Shape(v|n−1)μ} and writeAˇ^μfor theR–submodule ofS^λfreely generated by

{cv:v∈Std(λ)and Shape(v|n−1)μ}.

Ifv∈Stdn(λ)andv|n−1=t^μ, then theR–linear map determined on generators by c_vX_d(u)+ ˇA^μ→c_u, foru∈Std(μ),

is an isomorphismA^μ/Aˇ^μ∼=C^μofHn−1(q²)–modules.

The Jucys–Murphy operatorsD˜_iinHn(q²)are usually defined (Sect. 3 of [8]) by D˜1=0 and

D˜_i=

i−1

k=1

X_(k,i), fori=1, . . . , n. (2.9)

As per an exercise in [8], we define D₁=1 and set D_i =X_i−1D_i−1X_i−1. Since D_i=1+(q−q⁻¹)D˜_i, and theD˜_i can be cumbersome, we work with theD_irather than theD˜_i. We also refer to theD_i as Jucys–Murphy elements; this should cause no confusion. The following proposition is well known.

Proposition 2.4 Letiandkbe integers, 1≤i < nand 1≤k≤n.

1. Xi andDk commute ifi=k−1, k.

2. Di andDkcommute.

3. Xi commutes withDiDi+1andDi+Di+1.

Lett=(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ⁽ⁿ⁾)be a standardλ–tableau identified with the correspond- ing path in the Bratteli diagram ofHn(q²). For each integerkwith 1≤k≤n, define P_t(k)=q^2(j−i) where[λ^(k)] = [λ^(k−1)] ∪ {(i, j )}. (2.10) The next statement is due to R. Dipper and G. James (Theorem 3.32 of [8]).

Theorem 2.5 Suppose thatλis a partition ofnand letsbe a standardλ–tableau. If kis an integer, 1≤k≤n, then there existav∈R, forvs, such that

c_sD_k=P_s(k)c_s+

v∈Std(λ) vs

a_vc_v.

One objective at hand is to provide an extension of Lemma2.3and Theorem2.5to the Brauer and Birman–Murakami–Wenzl algebras.

3 The Birman–Murakami–Wenzl algebras

Letq, r be indeterminates overZandR=Z[q^±1, r^±1, (q−q⁻¹)⁻¹]. The Birman–

Murakami–Wenzl algebraB_n(q, r)overRis the unital associativeR–algebra gener- ated by the elementsT1, T2, . . . , Tn−1, which satisfy the defining relations

(Ti−q)(Ti+q⁻¹)(Ti−r⁻¹)=0 for 1≤i < n;

T_iT_i+1T_i =T_i+1T_iT_i+1 for 1≤i≤n−2;

T_iT_j=T_jT_i for 2≤ |i−j|;

E_iT_i^±₋¹₁E_i=r^±1E_i for 2≤i≤n−1;

E_iT_i^±₊¹₁E_i=r^±1E_i for 1≤i≤n−2;

TiEi=EiTi=r⁻¹Ei for 1≤i≤n−1, whereE_i is the element defined by the expression

(q−q⁻¹)(1−E_i)=T_i−T_i⁻¹. Writing

z=(q+r)(qr−1)

r(q+1)(q−1), (3.1)

then (Sect. 3 of [12]) one derives additional relations E_i²=zEi,

EiT_i^±1=r^∓1Ei=T_i^±1Ei, T_i²=1+(q−q⁻¹)(T_i−r⁻¹E_i) E_i±1T_iT_i±1=T_iT_i±1E_i

EiTi±1Ei=rEi

EiT_i⁻_±¹₁Ei=r⁻¹Ei

EiEi±1Ei=Ei

E_iE_i±1=E_iT_i±1T_i =T_i±1T_iE_i±1.

Ifw∈S_nis a permutation andw=s_i1s_i2· · ·s_ik is a reduced expression forw, then T_w=T_i1T_i2· · ·T_ik

is a well defined element ofBn(q, r).

Remark 3.1 The generatorT_i above differs by a factor ofq from the generator used in [4] but coincides with the elementg_i of [6] and [12].

Iff is an integer, 0≤f ≤ [n/2], defineB_n^f to be the two sided ideal ofBn(q, r) generated by the elementE₁E₃· · ·E_2f−1. Then

(0)⊆B_n^[n/2]⊆B_n^[n/2]−1⊆ · · · ⊆B_n¹⊆B_n⁰=B_n(q, r) (3.2) gives a filtration ofBn(q, r). As in Theorem 4.1 of [4] (see also [13]), refining the filtration (3.2) gives the cell modules, in the sense of [5], for the algebraB_n(q, r). If f is an integer, 0≤f ≤ [n/2], andλis a partition ofn−2f, define the element

x_λ=

w∈S_λ

q^(w)T_w,

whereSλ is row stabiliser in the subgroups_i :2f < i < n of the superstandard tableaut^λ∈Stdn(λ)as defined in Sect.2; finally define

m_λ=E1E3· · ·E2f−1x_λ

which is the analogue to the elementcλin the Iwahori-Hecke algebra of the symmet- ric group.

Example 3.1 Let n=10 and λ=(3,2,1). From the λ–tableau displayed in (2.2) comes the subgroupSλ= s₅, s₆, s₈, and

m_λ=E1E3

w∈Sλ

q^(w)T_w

=E₁E₃(1+qT₅)(1+qT₆+q²T₆T₅)(1+qT₈).

Iff is an integer, 0≤f ≤ [n/2], define Df,n=

⎧⎨

⎩v∈Sn

(2i+1)v < (2j+1)vfor 0≤i < j < f; (2i+1)v < (2i+2)vfor 0≤i < f;

and(i)v < (i+1)vfor 2f < i < n

⎫⎬

⎭.

As shown in Sect. 3 of [4], the collectionDf,nis a complete set of right coset repre- sentatives for the subgroupB_f ×S_n−2f inS_n, whereS_n−2f is identified with the subgroupsi:2f < i < nofSnandB0= 1,B1= s1and, forf >1,

B_f = s_2i−1, s_2is_2i+1s_2i−1s_2i:1≤i≤f.

Additionally, it is evident (Proposition 3.1 of [4]) that ifvis an element ofDf,n, then (uv)=(u)+(v)for alluinsi:2f < i < n.

Remark 3.2 After fixing a choice of over and under crossings, there is a natural bi- jection between the coset representativesDf,nand the(n, n−2f )–dangles of Defi- nition 3.3 of [13].

For each partitionλofn−2f, defineIn(λ)to be the set of ordered pairs In(λ)=

(s, v):s∈Stdn(λ)andv∈Df,n,

(3.3) and defineB_n^λto be the two–sided ideal inB_n(q, r)generated bym_λand let

Bˇ_n^λ=

μλ

B_n^μ

so thatB_n^f+1⊆ ˇB_n^λ, by the definition of the dominance order on partitions given in Sect.2. Let∗be the algebra anti–involution ofB_n(q, r)which mapsT_w→T_w−1 and E_i→E_i.

ThatB_n(q, r)is cellular in the sense of [5] was shown in [13]; the next statement which is Theorem 4.1 of [4], gives an explicit cellular basis forB_n(q, r).

Theorem 3.1 The algebraB_n(q, r)is freely generated as anR–module by the col- lection

T_v^∗T_d(s)^∗ m_λT_d(t)T_u

(s, v), (t, u)∈In(λ), forλa partition ofn−2f, and 0≤f ≤ [n/2]

.

Moreover, the following statements hold.

1. The algebra anti–involution∗satisfies

∗ :T_v^∗T_d(s)^∗ m_λT_d(t)T_u→T_u^∗T_d(t)^∗ m_λT_d(s)T_v for all(s, v), (t, u)∈In(λ).

2. Suppose thatb∈B_n(q, r)and letfbe an integer, 0≤f≤ [n/2]. Ifλis a partition ofn−2f and(t, u)∈In(λ), then there exista_(u,w)∈R, for(u, w)∈In(λ), such that for all(s, v)∈In(λ),

T_v^∗T_d(s)^∗ mλTd(t)Tub≡

(u,w)

a(u,w)T_v^∗T_d(s)^∗ mλTd(u)Tw modBˇ_n^λ. (3.4)

As a consequence of the above theorem,Bˇ_n^λis theR–module freely generated by the collection

T_v^∗T_d(s)^∗ m_μT_d(t)T_u:(s, v), (t, u)∈In(μ), forμλ .

Iff is an integer, 0≤f ≤ [n/2], andλis a partition ofn−2f, the cell moduleS^λ is defined to be theR–module freely generated by

m_λT_d(t)T_u+ ˇB_n^λ :(t, u)∈In(λ)

(3.5)

and given the rightB_n(q, r)action m_λT_d(t)T_ub+ ˇB_n^λ=

(u,w)

a_(u,w)m_λT_d(u)T_w+ ˇB_n^λ forb∈B_n(q, r), where the coefficientsa_(u,w)∈R, for(u, w)inIn(λ), are determined by the expres- sion (3.4).

Example 3.2 Letn=6,f =1, andλ=(3,1). Ifi, jare integers with 1≤i < j≤n, writev_i,j=s₂s₃· · ·s_j−1s₁s₂· · ·s_i−1, so that

Df,n= {vi,j:1≤i < j≤n}.

Since

Stdn(λ)=

t^λ= ,t^λs5= ,t^λs5s4=

andm_λ=E₁(1+qT₄)(1+qT₃+q²T₃T₄), the basis forS^λ, of the form displayed in (3.5), is

m_λT_d(s)T_vi,j+ ˇB_n^λ:s∈Stdn(λ)and 1≤i < j≤n .

As in Proposition 2.4 of [5], the cell moduleS^λforB_n(q, r)admits a symmetric associative bilinear form, :S^λ×S^λ→Rdefined by

mλTd(u)Tv, mλTd(v)Twmλ≡mλTd(u)TvT_w^∗T_d(v)^∗ mλ modBˇ_n^λ. (3.6) We return to the question of using the bilinear form (3.6) to extract explicit informa- tion about the structure of the B–W–W algebras in Sect.8, but record the following example for later reference.

Example 3.3 Let n=3 and λ=(1)so thatBˇ_n^λ=(0)andm_λ=E₁. We order the basis (3.5) for the module S^λ as v1=E₁, v2=E₁T₂ and v3=E₁T₂T₁ and, with respect to this ordered basis, the Gram matrixvi,vjof the bilinear form (3.6) is

⎡

⎣z r 1 r z+(q−q⁻¹)(r−r⁻¹) r⁻¹

1 r⁻¹ z

⎤

⎦.

The determinant of the Gram matrix given above is (r−1)²(r+1)²(q³+r)(q³r−1)

r³(q−1)³(q+1)³ . (3.7)

Remark 3.3 (i) Letκ be a field andr,ˆ q, (ˆ qˆ− ˆq⁻¹)be units inκ. The assignments ϕ:r→ ˆrandϕ:q→ ˆqdetermine a homomorphismR→κ, givingκanR–module structure. We refer to the specialisationBn(q,ˆ r)ˆ =Bn(q, r)⊗Rκ as a B–M–W al- gebra overκ. If 0≤f ≤ [n/2]andλis a partition ofn−2f then the cell module

S^λ⊗Rκ forB_n(q,ˆ r)ˆ admits a symmetric associative bilinear form which is related to the generic form (3.6) in an obvious way.

(ii) Whenever the context is clear and no possible confusion will arise, the abbre- viationS^λwill be used for theB_n(q,ˆ r)–moduleˆ S^λ⊗Rκ.

The proof of Theorem 3.1given in [4] rests upon the following facts, respectively Proposition 3.2 of [12] and Proposition 3.3 of [4], stated below for later reference.

Lemma 3.2 Let f be an integer, 0≤f ≤ [n/2], write C_f for the subalgebra of B_n(q, r)generated by the elementsT_2f+1, . . . , T_n−1, andI_f for the two sided ideal ofCf generated by the elementE2f+1. Then the map defined on algebra generators ofHn−2f(q²)by

φ:X_i→T_2f+i+I_f, for 1≤i < n−2f,

and extended to all ofHn−2f byφ (h₁h₂)=φ (h₁)φ (h₂)wheneverh₁, h₂∈Hn−2f, is an algebra isomorphismHn−2f(q²)∼=C_f/I_f.

Lemma 3.3 Letf be an integer, 0≤f <[n/2], andCf andIf be as in Lemma3.2 above. Ifiis an integer, 2f < i < n, andb∈C_f, then

E₁E₃· · ·E_2f−1bE_i≡E₁E₃· · ·E_2f−1E_ib≡0 modB_n^f+1.

Since Hn−2f(q²)⊆Hn(q²) is generated by {X_j :1≤j < n−2f}, from Lem- mas3.2and3.3we obtain Corollary3.4; cf. Sect. 3 of [4].

Corollary 3.4 If f is an integer, 0≤f <[n/2], then there is a well defined R–

module homomorphismϑf :Hn−2f(q²)→B_n^f/B_n^f+1, determined by ϑ_f :X_vˆ→E₁E₃· · ·E_2f−1T_v+B_n^f+1,

wherev=s_i1s_i2· · ·s_id is a permutation ins_i:2f < i < nandwˆ is the permutation ˆ

v=s_i1−2fs_i2−2f· · ·s_id−2f. Additionally, the mapϑ_f satisfies the property

ϑ_f(X_vˆX_j)=ϑ_f(X_vˆ)T2f+j, (3.8) whenever 1≤j < n−2f.

Remark 3.4 The fact thatϑf is an isomorphism ofR–modules was not used in the proof of Theorem3.1; however it may be deduced from Theorem3.1which implies that the dimension over R of the image space ofϑ_f is equal to the dimension of Hn−2f(q²)overR.

Lemma 3.5 Let f be an integer, 0< f ≤ [n/2]. If b∈B_n(q, r), w∈Df,n, and 1≤i < n, then there exista_u,vinR, foruins_i:2f < i < nandvinDf,n, uniquely determined by

E₁E₃· · ·E_2f−1T_wb≡

u,v

a_u,vE₁E₃· · ·E_2f−1T_uT_v modB_n^f+1. (3.9)

Proof For the uniqueness of the expression (3.9), observe that there is a one–to–one map

E₁E₃· · ·E_2f−1T_uT_v+B_n^f+1→

s,t∈Std_n(λ) λn−2f

a_s,tT_d(s)^∗ m_λT_d(t)T_v+B_n^f+1,

foru∈ s_j : 2f < j < nandv∈Df,n, determined by the mapϑ_f and the transition between the basis{Xw:w∈Sn−2f}and the Murphy basis forHn−2f(q²), where the expression on the right hand side above is anR–linear sum of the basis elements forBn^f/Bn^f+1given by Theorem3.1.

The proof of the lemma makes repeated use of the following fact. Ifu∈ s_i:2f <

i < nandv∈Sn, thenE1E3· · ·E2f−1T_uT_v is expressible as a sum of the form that appears on the right hand side of (3.9). To see this, first note that, given an integer iwith 2f < i < nand(i+1)v< (i)v,

T_uT_v=

T_us_iT_siv, if(u) < (us_i);

(T_usi+(q−q⁻¹)(T_u−r⁻¹T_usiE_i))T_siv, otherwise.

Thus, using Lemma3.3, we havea_u,v∈R, foru∈ s_i : 2f < i < n andv∈Sn, such that

E1E3· · ·E2f−1T_uT_v≡

u,v

au,vE1E3· · ·E2f−1TuTv modB_n^f+1, where (i)v < (i+1)v, for 2f < i < n, whenever a_u,v=0 in the above expres- sion. Noting thatE1E3· · ·E2f−1Tv=r⁻¹E1E3· · ·E2f−1Ts_2i−1v if 1≤i≤f and (s_2i−1v) < (v), and applying Proposition 3.7 or Corollary 3.1 of [4], we may as- sume thatv∈D_f,n, whenevera_u,v=0 in the above expression.

Proceeding with the proof of the lemma, first consider the case whereb=E_i for some 1≤i < n. Letk=(i)w⁻¹andl=(i+1)w⁻¹. If(i+1)w⁻¹< (i)w⁻¹, then T_wE_i =r⁻¹T_wsiE_i, wherews_i∈Df,n. We may therefore suppose thatk < l. Using Proposition 3.4 of [4],

T_wE_i=

E_kT_w, ifl=k+1;

T_l^ε₋^l−₁¹T_l^ε₋^l−₂²· · ·T_k^ε₊^k+₁¹EkT_w, otherwise, (3.10) wherew=s_k+1s_k+2· · ·s_l−1wand, fork < j < l,

ε_j=

1, ifi+1< (j )w;

−1, otherwise.

Considering the two cases in (3.10) separately, multiply both sides of the expres- sion (3.10) byE₁E₃· · ·E_2f−1. Ifl=k+1, then

E1E3· · ·E2f−1TwEi=

⎧⎪

⎨

⎪⎩

zE1E3· · ·E2f−1Tw, ifk <2f andkis odd;

E₁E₃· · ·E_2f−1T_kT_k−1T_w, ifk≤2f andkis even;

E1E3· · ·E2f−1EkTw, if 2f < k.

(3.11)