Research Article | Open Access
The Structure of Simple Modules of Birman-Murakami-Wenzl Algebras
We study the restriction of simple modules of Birman-Murakami-Wenzl algebras with being not a root of 1. Precisely, we study the module structure for the restriction of to and describe the socle and head of the restriction of each simple module completely.
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 and orthogonal or symplectic groups.
Birman-Wenzl  and Murakami  introduced a class of associative algebras independently, called Birman-Murakami-Wenzl algebras, in order to study link invariants. On the other hand, there is a Schur-Weyl duality between with some special parameters over and quantum groups of types B, C, and D . So, BMW algebras can be seen as the -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 to for simple modules in Section 3.
2. Birman-Murakami-Wenzl Algebra
In this section, we recall some results on the BMW algebra  over an integral domain , where and , are indeterminates.
Definition 1 (see ). The BMW algebra is a unital associative -algebra generated by , subject to the following relations: (a), for ,(b)(1) if , (2) , for ,(c) (1) , for , (2) , for and ,where for .
It is well-known that the Hecke algebra associated with the symmetric group is a quotient algebra of BMW algebra . Morton and Wassermann  proved that is free over with rank .
When is semisimple, the simple modules are cell modules in the sense of . The branching rule of cell modules is well-known . However, the algebra 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 ; BMW algebra may not be semisimple even if is not a root of . In this paper, we study the restriction of simple modules of BMW algebra over a field and with the assumption that is not a root of . According to , we will assume for some and . Otherwise, is semisimple.
Now, we need some of combinatorics in order to state the results on .
Let , where is the set of partitions of . By definition, each is a weakly decreasing sequence of nonnegative integers such that , the summation of those integers, is . For , we say that dominates and write , if in the usual sense or and in the sense that for all . So, is a poset. We write if for all possible . Let and .
The Young diagram for a partition is a collection of boxes with boxes in the th row of . A box of is called removable (resp., addable) if it can be removed from (resp., added to) such that the result is still a Young diagram for a partition. A -tableau is obtained by inserting , , into without repetition. We denote the set of all removable boxes of by and the set of all addable boxes of by .
We recall the definition of a cellular algebra in .
Definition 2 (see ). Let be a commutative ring and an -algebra. Fix a partially ordered set . Then for each , let be a finite set and , where . The triple is a cell datum for if(a) and is an -basis for ;(b)there is an -linear anti-involution on such that , for all and all ;(c)we let -span and . For any , , and there exist scalars such that Algebra is a cellular algebra if it has a cell datum.
Xi  proved that is a cellular algebra over associated with the poset .
We recall the representations of over a field as follows. For each , there is a cell module for . On each , there is an invariant form, say . Let be the radical of . Then the corresponding quotient module is either zero or absolutely irreducible. In the latter case, we write it as . Let be the projective cover of .
Let mod- be the category of right -modules. We have embedding of the BMW algebras So, we have corresponding induction functor and restriction functor . Note that is exact functor and is right exact functor.
By standard arguments in [12, Section ], Rui and Si defined the exact functor and right exact functor in , which satisfyfor all and . For the simplification of notation, we will use and instead of and , 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 [14, Lemma ]). Suppose that and . We have(1),(2),(3),(4) as vector spaces.
At the end of this section, we recall the branching rule for cell modules of over .
Theorem 4 (see ). For each , then has a filtration of -modules such that , where ranges over all partitions obtained from by either removing a removable node if or adding an addable node if . Further, the multiplicity of in is one. In particular, this result is true over an arbitrary field.
Notation. If appear in the section defined by the above theorem, we write .
In , the author proved the following result for cyclotomic BMW algebra and so for BMW algebra.
Lemma 5. .
3. The Structure of Simple Modules of
In this section, we describe the module structure of the restriction of to for .
For each partition and each , define by(1) if ,(2) if ,(3) if ,(4) if , where is the conjugate of .
It is easy to see that is a strictly decreasing sequence. Let (or for brevity) be the number of pairs in .
Example 7. Assume and ; then and . The pairs in are and , so .
Let be the set of all infinite sequences such that . Define to be the reflection group on generated by the reflections and , where Rui and Si classified the blocks of BMW algebras with . Based on this result, it was shown in  that two simple -modules and are in the same block if and only if . For , let be the block containing . Now we define to be the set of partitions in the -orbit of . So we have where the union is taken over all such that for some .
We consider the projection functor from the category of -module to the block of which contains . So we have By Theorem 4, we know that the direct sum can be taken over all blocks with , where , for some .
According to the definition of , it is easy to see that there are three cases to consider depending on the relation between and with :(1),(2),(3).
Now, we need some notation in order to state the result of case (1).
Definition 8 (see ). Two partitions and are said to be translation equivalent if (a) and ;(b)for each , there is unique such that and .
Proposition 9 (see ). Let , , and . If , then and are translation equivalent.
Theorem 10. Let , , and . If , then
Proof. According to the theory of cellular algebra, we have an epimorphism . Applying the functor to , we have an epimorphism by Theorem 4 and Proposition 9. Hence has simple head .
If is in the socle of , then must be a composition factor of . So, we have The last equality follows from Definition 8 and Proposition 9.
Since , we have . Hence we have .
So has simple head and simple socle . However, the composition factors of must be the composition factors of and , so we have .
In order to deal with case (2) and case (3), we need some notation here.
Definition 11 (see ). Suppose that . We say separates and if with one of or being equal to and (1) is the unique element of ;(2) is the unique element of ;(3) and are the unique pair of elements of .
For (or ), we denote the partition corresponding to the Young diagram (or ) by (or by ).
Proposition 12. Let , , and . If , then separates and for some or separates and for some .
Remark 13. Fixing the above notation, we assume that . So, we write and instead of and and write and instead of and , respectively.
Theorem 14. Let , , and be as above. Then we have (1),(2).
Proof. Similar to the proof of Theorem 10, we have two epimorphisms and . Applying the functor to and , we have two epimorphisms and by Theorem 4 and Proposition 12. Hence and are either 0 or have simple head .
By [9, Corollary 5.8] and Proposition 12, we have an exact sequence So, we have Hence has simple head .
With the same reason as Theorem 10, we complete the proof of (1).
Since , according to the proof of (1), the unique copy of must come from . So cannot have simple head ; this implies that .
Keeping the same notation, for case (3), we need to describe for and . In this paper, we only describe the head and socle of .
Theorem 15. Let and be as above. Then (1)if , then ;(2)if , then has simple head and simple socle .
Proof. When , can be considered as the cell module of Hecke algebra . However, under our assumption, Hecke algebra is semisimple. So we have . Similarly, we have . So, (1) follows from Theorem 4 and Proposition 12.
Similar to the proof of Theorem 10, we have an epimorphism If is in the head of , it must be in the head of .
By [9, Corollary 5.8] and Proposition 12, we have an exact sequenceIt is easy to see that is in the head of . Note that So, we have The last equality follows from Theorem 14(2).
Hence has simple head . It follows that has simple head .
Assume that is in the socle of . Then Hence the socle of consists of and by Proposition 12.
Suppose that is in the socle of . Consider the set is a composition factor of . According to the assumption, and must be the maximal one in the poset ; hence must come from . So, we have .
Hence we have the following commutative diagram:It follows that . This implies that . With the exact sequence (12), we have the following commutative diagram:Since and are two epimorphisms, is an epimorphism. This means that the composition factors of must be composition factors of except one copy of .
However, we have The last equality follows from the proof of [6, Proposition 4.13].
It is in contradiction with . Hence cannot be in the socle of . It follows that is in the socle of . Since we have proved that has simple head , has simple socle .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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