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 [1] 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 [2] and Murakami [3] 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 [4]. 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 [5] described the module structure of the restriction of simple modules of Brauer algebras by using certain combinatorial graph. In [6], 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) [2] over an integral domain R≔Z[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, 1≤i<n subject to the following relations:

(Ti-q)(Ti+q-1)(Ti-r-1)=0, for 1≤i<n,

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

(2) TiTi+1Ti=Ti+1TiTi+1, for 1≤i<n-1,

(1) EiTi=r-1Ei=TiEi, for 1≤i≤n-1,

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

where Ei=1-ω-1(Ti-Ti-1) for 1≤i≤n-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 [7] 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 [8]. The branching rule of cell modules is well-known [9]. 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 [10].

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 [10], we will assume r=ɛqa for some a∈Z 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,λ∣0≤f≤n/2,λ∈Λ+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 λ⊵μ in the sense that (1)∑k=1lλk≥∑k=1lμkfor all l≥1. So, Λn is a poset. We write λ⊇μ if λj≥μj for all possible j. Let Λ=⋃n=1∞Λn and Λ+=⋃n=1∞Λ+(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, 1≤i≤n, 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 [8].

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 λ∈Λ, let T(λ) be a finite set and Cstλ∈A, where s,t∈T(λ). The triple (Λ,T,C) is a cell datum for A if

{Cstλ∣λ∈Λ and s,t∈T(λ)} is an R-basis for A;

there is an R-linear anti-involution ∗ on A such that (Cstλ)∗=Ctsλ, for all λ∈Λ and all s,t∈T(λ);

we let A⊳λ=R-span {Cuvμ∣μ⊳λ and u,v∈T(μ)}. For any λ∈Λ, s∈T(λ), and a∈A there exist scalars rtu(a)∈R such that (2)Cstλa=∑u∈TλrtuaCsuλmodA⊳λ.

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

Xi [11] 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,q↪Bnr,q↪Bn+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 [13], which satisfy(4)FnM=MEn-1,Gn-2N=N⨂Bn-2r,qEn-1Bnr,qfor all M∈mod-Bn(r,q) and N∈mod-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 [14] 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=M0⊆M1⊆⋯⊆Mm=Δ(f,λ) of Bn-1(r,q)-modules such that Mi/Mi-1≅Δ(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 [6], the author proved the following result for cyclotomic BMW algebra and so for BMW algebra.

Lemma 5.

resn+2?∘Gn?=indn?.

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 a∈Z, 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 vi∈R. Define W to be the reflection group on X generated by the reflections (i,j) and (i,j)-(i,j∈Z>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 [13]. Based on this result, it was shown in [6] 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=⨁BμProjμ∘resnDf,λ. By Theorem 4, we know that the direct sum can be taken over all blocks B(μ) with μ∈supp(λ), where supp(λ)={μ∈Λ+∣(l,μ)→(f,λ), for some f,l∈Z≥0}.

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 μ∈supp(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 λ′∈B(λ), there is unique μ′∈B(μ) such that Bλ∩suppμ′=λ′ and Bμ∩suppλ′=μ′.

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

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

Theorem 10.

Let (f,λ)∈Λn, (l,μ)∈Λn-1, and μ∈supp(λ). If s(λ)=s(μ), then (7)Projμ∘resnDf,λ=Dl,μ.

Proof.

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

If Dk,ν is in the socle of Projμ∘resnDf,λ, then Dk,ν must be a composition factor of Δ(l,μ). So, we have (8)0≠HomΔk,ν,Projμ∘resnDf,λ≅HomΔk,ν,resnDf,λ≅Homindn-1Δk,ν,Df,λ≅HomProjλ∘indn-1Δk,ν,Df,λ=HomΔk′,ν′,Df,λ. The last equality follows from Definition 8 and Proposition 9.

Since HomΔk′,ν′,Df,λ≠0, we have ν′=λ. Hence we have ν=μ.

So Projμ∘resnDf,λ has simple head Dl,μ and simple socle Dl,μ. However, the composition factors of Projμ∘resnDf,λ must be the composition factors of Δ(l,μ) and [Δ(l,μ):Dl,μ]=1, so we have Projμ∘resnDf,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 λ′∈supp(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 p∈R(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 λ′∈supp(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 λ′∈supp(λ+) be as above. Then we have

Projλ′∘resnDf,λ+=Df,λ′,

Projλ′∘resnDf+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 Projλ′∘resn to ψ+ and ψ-, we have two epimorphisms Δ(f,λ′)→Projλ′∘resnDf,λ+ and Δ(f,λ′)→Projλ′∘resnDf+1,λ- by Theorem 4 and Proposition 12. Hence Projλ′∘resnDf,λ+ and Projλ′∘resnDf+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)0⟶Δf+1,λ-⟶Projλ+∘indn-1Δf,λ′⟶Δf,λ+⟶0. So, we have (10)HomΔf,λ′,Projλ′∘resnDf,λ+≅HomΔf,λ′,resnDf,λ+≅Homindn-1Δf,λ′,Df,λ+≅HomProjλ+∘indn-1Δf,λ′,Df,λ+≠0.

Hence Projλ′∘resnDf,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 Projλ′∘resnDf,λ+. So Projλ′∘resnDf+1,λ- cannot have simple head Df,λ′; this implies that Projλ′∘resnDf+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,λ′⟶Projλ+∘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)0⟶Δf,λ-⟶Projλ+∘resnΔf,λ′⟶Δf-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′,Projλ′∘resn-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)0≠HomΔl,μ,Projλ+∘resnDf,λ′≅HomΔl,μ,resnDf,λ′≅Homindn-1Δl,μ,Df,λ′≅HomProjλ′∘indn-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-1∣Dl,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-=kerϕ′⊕Df,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,λ′≅dim HomProjλ′∘indn-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.

BrauerR.On algebras which are connected with the semisimple continuous groupsBirmanJ. S.WenzlH.Braids, link polynomials and a new algebraMurakamiJ.The Kauffman polynomial of links and representation theoryKirillovA. N.ReshetihikhinY. N.KacV. G.Representations of the algebras Uqsl2, q-orthogonal polynomials and invariants of linksDe VisscherM.MartinP.On Brauer algebra simple modules over the complex fieldXuX.Decomposition numbers of cyclotomic NW and BMW algebrasMortonH. R.A basis for the Birman-Wenzl algebrahttp://arxiv.org/abs/1012.3116GrahamJ. J.LehrerG. I.Cellular algebrasEnyangJ.Specht modules and semisimplicity criteria for Brauer and BIRman-Murakami-Wenzl algebrasRuiH.SiM.Gram determinants and semisimplicity criteria for Birman-Wenzl algebrasXiC. C.On the quasi-heredity of Birman-Wenzl algebrasGreenJ. A.SchockerM.ErdmannK.RuiH.SiM.Blocks of Birman-Murakami-Wenzl algebrasRuiH.SiM.The representations of cyclotomic BMW algebras, IICoxA.De VisscherM.MartinP.Alcove geometry and a translation principle for the Brauer algebra