JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 10.1155/2015/876251 876251 Research Article The Structure of Simple Modules of Birman-Murakami-Wenzl Algebras Xu Xu Jing Naihuan School of Mathematics and Statistics Wuhan University Wuhan 430072 China whu.edu.cn 2015 12102015 2015 06 07 2015 16 09 2015 20 09 2015 12102015 2015 Copyright © 2015 Xu Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the restriction of simple modules Df,λ of Birman-Murakami-Wenzl algebras Bn(r,q) with q  being not a root of 1. Precisely, we study the module structure for the restriction of Df,λ to Bn-1(r,q) and describe the socle and head of the restriction of each simple module completely.

1. Introduction

Classical Schur-Weyl duality relates the representations of the symmetric and general linear groups via their actions on tensor space. Brauer  introduced a class of algebras called Brauer algebras to generalize the classical Schur-Weyl duality. He proved that there is a Schur-Weyl duality between Brauer algebras with some special parameters over C and orthogonal or symplectic groups.

Birman-Wenzl  and Murakami  introduced a class of associative algebras Bn(r,q) independently, called Birman-Murakami-Wenzl algebras, in order to study link invariants. On the other hand, there is a Schur-Weyl duality between Bn(r,q) with some special parameters over C and quantum groups of types B, C, and D . So, BMW algebras can be seen as the q-deformation of Brauer algebras.

Recently, based on the results of decomposition numbers of Brauer algebras, De Visscher and Martin  described the module structure of the restriction of simple modules of Brauer algebras by using certain combinatorial graph. In , the author gave a combinatorial algorithm for computing decomposition numbers of BMW algebras. Motivated by these works, we study the restriction of simple modules of BMW algebras in this paper.

We organize this paper as follows: in Section 2, we recall some results on representation theory of BMW algebras. Then, we will describe the structure of the restriction from Bn(r,q) to Bn-1(r,q) for simple modules in Section 3.

2. Birman-Murakami-Wenzl Algebra

In this section, we recall some results on the BMW algebra Bn(r,q)  over an integral domain RZ[r±1,q±1,ω-1], where ω=q-q-1 and q, r are indeterminates.

Definition 1 (see [<xref ref-type="bibr" rid="B1">2</xref>]).

The BMW algebra Bn(r,q) is a unital associative R-algebra generated by Ti, 1i<n subject to the following relations:

(Ti-q)(Ti+q-1)(Ti-r-1)=0, for 1i<n,

(1) TiTj=TjTi if |i-j|>1,

(2) TiTi+1Ti=Ti+1TiTi+1, for 1i<n-1,

(1) EiTi=r-1Ei=TiEi, for 1in-1,

(2) EiTj±1Ei=r±1Ei, for 1in-1 and j=i±1,

where Ei=1-ω-1(Ti-Ti-1) for 1in-1.

It is well-known that the Hecke algebra Hn associated with the symmetric group Sn is a quotient algebra of BMW algebra Bn(r,q). Morton and Wassermann  proved that Bn(r,q) is free over R with rank (2n-1)!!.

When Bn(r,q) is semisimple, the simple modules are cell modules in the sense of . The branching rule of cell modules is well-known . However, the algebra Bn(r,q) is not always semisimple. In 2009, Rui and Si gave a necessary and sufficient condition for BMW algebras being semisimple over an arbitrary field .

It is different from Hecke algebra Hn; BMW algebra Bn(r,q) may not be semisimple even if q is not a root of 1. In this paper, we study the restriction of simple modules of BMW algebra Bn(r,q) over a field and with the assumption that q is not a root of 1. According to , we will assume r=ɛqa for some aZ and ɛ=±1. Otherwise, Bn(r,q) is semisimple.

Now, we need some of combinatorics in order to state the results on Bn(r,q).

Let Λn=f,λ0fn/2,λΛ+n-2f, where Λ+(n-2f) is the set of partitions of n-2f. By definition, each λ=(λ1,λ2,)Λ+(n-2f) is a weakly decreasing sequence of nonnegative integers such that |λ|, the summation of those integers, is n-2f. For (f,λ),(l,μ)Λn, we say that (f,λ) dominates (l,μ) and write (f,λ)(l,μ), if f>l in the usual sense or f=l and λμ in the sense that (1)k=1lλkk=1lμkfor all l1. So, Λn is a poset. We write λμ if λjμj for all possible j. Let Λ=n=1Λn and Λ+=n=1Λ+(n).

The Young diagram Y(λ) for a partition λ=(λ1,λ2,) is a collection of boxes with λi boxes in the ith row of Y(λ). A box of Y(λ) is called removable (resp., addable) if it can be removed from (resp., added to) Y(λ) such that the result is still a Young diagram for a partition. A λ-tableau s is obtained by inserting i,  1in, into Y(λ) without repetition. We denote the set of all removable boxes of Y(λ) by R(λ) and the set of all addable boxes of Y(λ) by A(λ).

We recall the definition of a cellular algebra in .

Definition 2 (see [<xref ref-type="bibr" rid="B6">8</xref>]).

Let R be a commutative ring and A an R-algebra. Fix a partially ordered set Λ=(Λ,). Then for each λΛ, let T(λ) be a finite set and CstλA, where s,tT(λ). The triple (Λ,T,C) is a cell datum for A if

{CstλλΛ and s,tT(λ)} is an R-basis for A;

there is an R-linear anti-involution on A such that (Cstλ)=Ctsλ, for all λΛ and all s,tT(λ);

we let Aλ=R-span {Cuvμμλ and u,vT(μ)}. For any λΛ, sT(λ), and aA there exist scalars rtu(a)R such that (2)Cstλa=uTλrtuaCsuλmodAλ.

Algebra A is a cellular algebra if it has a cell datum.

Xi  proved that Bn(r,q) is a cellular algebra over R associated with the poset Λn.

We recall the representations of Bn(r,q) over a field as follows. For each (f,λ)Λn, there is a cell module Δ(f,λ) for Bn(r,q). On each Δ(f,λ), there is an invariant form, say ϕf,λ. Let RadΔ(f,λ) be the radical of ϕf,λ. Then the corresponding quotient module Δ(f,λ)/RadΔ(f,λ) is either zero or absolutely irreducible. In the latter case, we write it as Df,λ. Let P(f,λ) be the projective cover of Df,λ.

Let mod-Bn(r,q) be the category of right Bn(r,q)-modules. We have embedding of the BMW algebras (3)Bn-1r,qBnr,qBn+1r,q. So, we have corresponding induction functor indn:mod-Bn(r,q)mod-Bn+1(r,q) and restriction functor resn:mod-Bn(r,q)mod-Bn-1(r,q). Note that resn is exact functor and indn is right exact functor.

By standard arguments in [12, Section 6], Rui and Si defined the exact functor Fn:mod-Bn(r,q)mod-Bn-2(r,q) and right exact functor Gn-2:mod-Bn-2(r,q)mod-Bn(r,q) in , which satisfy(4)FnM=MEn-1,Gn-2N=NBn-2r,qEn-1Bnr,qfor all Mmod-Bn(r,q) and Nmod-Bn-2(r,q). For the simplification of notation, we will use F and G instead of Fn and Gn, respectively.

The following results were proved by Rui and Si for cyclotomic BMW algebras  and we only need the special case here.

Lemma 3 (see [<xref ref-type="bibr" rid="B12">14</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M153"><mml:mrow><mml:mn>5.1</mml:mn></mml:mrow></mml:math></inline-formula>]).

Suppose that (f,λ)Λn and (l,μ)Λn+2. We have

FG=1,

G(Δ(f,λ))=Δ(f+1,λ),

F(Δ(f,λ))=Δ(f-1,λ),

HomBn+2(r,q)(G(Δ(f,λ)),Δ(l,μ))HomBn(r,q)(Δ(f,λ),F(Δ(l,μ))) as vector spaces.

At the end of this section, we recall the branching rule for cell modules of Bn(r,q) over R.

Theorem 4 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

For each (f,λ)Λn, then Δ(f,λ) has a filtration 0=M0M1Mm=Δ(f,λ) of Bn-1(r,q)-modules such that Mi/Mi-1Δ(l,μ), where μ ranges over all partitions obtained from λ by either removing a removable node if l=f or adding an addable node if l=f-1. Further, the multiplicity of Δ(l,μ) in resnΔ(f,λ) is one. In particular, this result is true over an arbitrary field.

Notation. If Δ(l,μ) appear in the section defined by the above theorem, we write (l,μ)(f,λ).

In , the author proved the following result for cyclotomic BMW algebra and so for BMW algebra.

Lemma 5.

r e s n + 2 ? G n ? = i n d n ? .

Remark 6.

Theorem 4, together with Lemmas 3 and 5, implies that there is a result for indnΔ(f,λ) similar to Theorem 4. Hence we will use Theorem 4 for indnΔ(f,λ) with no additional comments.

3. The Structure of Simple Modules of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M178"><mml:msub><mml:mrow><mml:mi mathvariant="script">B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo mathvariant="bold">(</mml:mo><mml:mi>r</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>q</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

In this section, we describe the module structure of the restriction of Df,λ to Bn-1(r,q) for (f,λ)Λn.

For each partition λ=(λ1,λ2,) and each aZ, define ρa,ea(λ)RN by

ρa=(-a+1/2,-a+3/2,) if r=qa,

ρa=(a-1/2,a-3/2,) if r=-qa,

ea(λ)=λt+ρa if r=qa,

ea(λ)=λ+ρa if r=-qa,

where λt is the conjugate of λ.

It is easy to see that ea(λ) is a strictly decreasing sequence. Let s(ea(λ)) (or s(λ) for brevity) be the number of pairs {p,-p} in ea(λ).

Example 7.

Assume r=-q3 and λ=(3,2,1); then ρa=(1,0,-1,-2,) and ea(λ)=(4,2,0,-2,-3,-4,-5,). The pairs {p,-p} in ea(λ) are {4,-4} and {2,-2}, so s(ea(λ))=2.

Let X be the set of all infinite sequences v=(v1,v2,,vk,) such that viR. Define W to be the reflection group on X generated by the reflections (i,j) and (i,j)-  (i,jZ>0), where (5)i,jv1,,vi,,vj,=v1,,vj,,vi,,i,j-v1,,vi,,vj,=v1,,-vj,,-vi,. Rui and Si classified the blocks of BMW algebras with o(q)>n . Based on this result, it was shown in  that two simple Bn(r,q)-modules Df,λ and Dl,μ are in the same block if and only if ea(λ)Wea(μ). For (f,λ)Λn, let Bn(λ) be the block containing λ. Now we define B(λ) to be the set of partitions in the W-orbit of λ. So we have B(λ)=mBm(λ) where the union is taken over all m such that (l,λ)Λm for some l.

We consider the projection functor Projλ from the category of Bn(r,q)-module to the block of Bn(r,q) which contains Δ(f,λ). So we have (6)resnDf,λ=BμProjμresnDf,λ. By Theorem 4, we know that the direct sum can be taken over all blocks B(μ) with μsupp(λ), where supp(λ)={μΛ+(l,μ)(f,λ), for some f,lZ0}.

According to the definition of s(λ), it is easy to see that there are three cases to consider depending on the relation between s(λ) and s(μ) with μsupp(λ):

s(λ)=s(μ),

s(λ)=s(μ)-1,

s(λ)=s(μ)+1.

Now, we need some notation in order to state the result of case (1).

Definition 8 (see [<xref ref-type="bibr" rid="B3">15</xref>]).

Two partitions λ and μ are said to be translation equivalent if

B(λ)supp(μ)={λ} and B(μ)supp(λ)={μ};

for each λB(λ), there is unique μB(μ) such that Bλsuppμ=λ and Bμsuppλ=μ.

Proposition 9 (see [<xref ref-type="bibr" rid="B15">6</xref>]).

Let (f,λ)Λn, (l,μ)Λn-1, and μsupp(λ). If s(λ)=s(μ), then λ and μ are translation equivalent.

Theorem 10.

Let (f,λ)Λn, (l,μ)Λn-1, and μsupp(λ). If s(λ)=s(μ), then (7)ProjμresnDf,λ=Dl,μ.

Proof.

According to the theory of cellular algebra, we have an epimorphism ψ:Δ(f,λ)Df,λ. Applying the functor Projμresn to ψ, we have an epimorphism Δ(l,μ)ProjμresnDf,λ by Theorem 4 and Proposition 9. Hence ProjμresnDf,λ has simple head Dl,μ.

If Dk,ν is in the socle of ProjμresnDf,λ, then Dk,ν must be a composition factor of Δ(l,μ). So, we have (8)0HomΔk,ν,ProjμresnDf,λHomΔk,ν,resnDf,λHomindn-1Δk,ν,Df,λHomProjλindn-1Δk,ν,Df,λ=HomΔk,ν,Df,λ. The last equality follows from Definition 8 and Proposition 9.

Since HomΔk,ν,Df,λ0, we have ν=λ. Hence we have ν=μ.

So ProjμresnDf,λ has simple head Dl,μ and simple socle Dl,μ. However, the composition factors of ProjμresnDf,λ must be the composition factors of Δ(l,μ) and [Δ(l,μ):Dl,μ]=1, so we have ProjμresnDf,λ=Dl,μ.

In order to deal with case (2) and case (3), we need some notation here.

Definition 11 (see [<xref ref-type="bibr" rid="B3">15</xref>]).

Suppose that λsupp(λ). We say λ separates λ- and λ+ if λ-λ+ with one of λ+ or λ- being equal to λ and

λ is the unique element of B(λ)supp(λ-);

λ is the unique element of B(λ)supp(λ+);

λ- and λ+ are the unique pair of elements of Bλ-suppλ.

For pR(λ) (or A(λ)), we denote the partition corresponding to the Young diagram Y(λ)p (or Y(λ)p) by λ-p (or by λ+p).

Proposition 12.

Let (f,λ)Λn, (l,λ)Λn-1, and λsupp(λ). If s(λ)=s(λ)-1, then λ separates λ and λ-p-p~ for some p,p~R(λ) or λ separates λ and λ+p+p~ for some p,p~A(λ).

Proof.

The first statement is [6, Proposition 4.11]. For the second statement, it is just needed to replace λ and λ-p-p~ by λ+p+p~ and λ in the proof of [6, Proposition 4.11], respectively.

Remark 13.

Fixing the above notation, we assume that λ+λ-. So, we write λ+ and λ- instead of λ and λ-p-p~ and write λ+ and λ- instead of λ+p+p~ and λ, respectively.

Theorem 14.

Let (f,λ+),(f+1,λ-)Λn, (f,λ)Λn-1, and λsupp(λ+) be as above. Then we have

ProjλresnDf,λ+=Df,λ,

ProjλresnDf+1,λ-=0.

Proof.

Similar to the proof of Theorem 10, we have two epimorphisms ψ+:Δ(f,λ+)Df,λ+ and ψ-:Δ(f+1,λ-)Df+1,λ-. Applying the functor Projλresn to ψ+ and ψ-, we have two epimorphisms Δ(f,λ)ProjλresnDf,λ+ and Δ(f,λ)ProjλresnDf+1,λ- by Theorem 4 and Proposition 12. Hence ProjλresnDf,λ+ and ProjλresnDf+1,λ- are either 0 or have simple head Dl,λ.

By [9, Corollary 5.8] and Proposition 12, we have an exact sequence (9)0Δf+1,λ-Projλ+indn-1Δf,λΔf,λ+0. So, we have (10)HomΔf,λ,ProjλresnDf,λ+HomΔf,λ,resnDf,λ+Homindn-1Δf,λ,Df,λ+HomProjλ+indn-1Δf,λ,Df,λ+0.

Hence ProjλresnDf,λ+ has simple head Df,λ.

With the same reason as Theorem 10, we complete the proof of (1).

Since [Δ(f,λ):Df,λ]=1, according to the proof of (1), the unique copy of Df,λ must come from ProjλresnDf,λ+. So ProjλresnDf+1,λ- cannot have simple head Df,λ; this implies that ProjλresnDf+1,λ-=0.

Keeping the same notation, for case (3), we need to describe Projλ+resnDf,λ for (f,λ)Λn and (f-1,λ+),(f,λ-)Λn-1. In this paper, we only describe the head and socle of Projλ+resnDf,λ.

Theorem 15.

Let (f,λ)Λn and (f-1,λ+),(f,λ-)Λn-1 be as above. Then

if f=0, then Projλ+resnD0,λ=D0,λ-;

if f>0, then Projλ+resnDf,λ has simple head Df-1,λ+ and simple socle Df-1,λ+.

Proof.

When f=0, Δ(0,λ) can be considered as the cell module of Hecke algebra Hn. However, under our assumption, Hecke algebra Hn is semisimple. So we have D0,λ=Δ(0,λ). Similarly, we have D0,λ-=Δ(0,λ-). So, (1) follows from Theorem 4 and Proposition 12.

Similar to the proof of Theorem 10, we have an epimorphism (11)ϕ:Projλ+resnΔf,λProjλ+resnDf,λ.

If Dl,μ is in the head of Projλ+resnDf,λ, it must be in the head of Projλ+resnΔ(f,λ).

By [9, Corollary 5.8] and Proposition 12, we have an exact sequence(12)0Δf,λ-Projλ+resnΔf,λΔf-1,λ+0.

It is easy to see that Df-1,λ+ is in the head of Projλ+resnΔ(f,λ). Note that (13)Projλ+resnΔf,λ=Projλ+indn-2Δf-1,λ.

So, we have (14)HomProjλ+resnΔf,λ,Df,λ-HomProjλ+indn-2Δf-1,λ,Df,λ-HomΔf-1,λ,Projλresn-1Df,λ-=0.

The last equality follows from Theorem 14(2).

Hence Projλ+resnΔ(f,λ) has simple head Df-1,λ+. It follows that Projλ+resnDf,λ has simple head Df-1,λ+.

Assume that Dl,μ is in the socle of Projλ+resnDf,λ. Then (15)0HomΔl,μ,Projλ+resnDf,λHomΔl,μ,resnDf,λHomindn-1Δl,μ,Df,λHomProjλindn-1Δl,μ,Df,λ.

Hence the socle of Projλ+resnDf,λ consists of Df-1,λ+ and Df,λ- by Proposition 12.

Suppose that Df,λ- is in the socle of Projλ+resnDf,λ. Consider the set Λn-1~={(l,μ)Λn-1Dl,μ is a composition factor of Projλ+resnDf,λ}. According to the assumption, (f,λ-)Λn-1~ and must be the maximal one in the poset Λn-1~; hence Df,λ- must come from Δ(f,λ-). So, we have ϕ(Δ(f,λ-))Df,λ-.

Hence we have the following commutative diagram:

It follows that ϕΔf,λ-=kerϕDf,λ-. This implies that ϕ(Δ(f,λ-))=Df,λ-. With the exact sequence (12), we have the following commutative diagram:

Since ϕ and π are two epimorphisms, η is an epimorphism. This means that the composition factors of Projλ+resnDf,λ must be composition factors of Δ(f-1,λ+) except one copy of Df,λ-.

However, we have (18)Projλ+resnDf,λ:Df-1,λ+=dim HomPf-1,λ+,Projλ+resnDf,λdim HomProjλindn-1Pf-1,λ+,Df,λ=2. The last equality follows from the proof of [6, Proposition 4.13].

It is in contradiction with [Δ(f-1,λ+):Df-1,λ+]=1. Hence Df,λ- cannot be in the socle of Projλ+resnDf,λ. It follows that Df-1,λ+ is in the socle of Projλ+resnDf,λ. Since we have proved that Projλ+resnDf,λ has simple head Df-1,λ+, Projλ+resnDf,λ has simple socle Df-1,λ+.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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