Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix B

B6. The Brauer algebras Bk(x)

Fix x. A Brauer diagram on k dots is a graph on two rows of k-vertices, one above the other, and k edges such that each vertex is incident to precisely one edge. The product of two k-diagrams d1 and d2 is obtained by placing d1 above d2 and identifying the vertices in the bottom row of d1 with the corresponding vertices in the top row of d2. The resulting graph contains k paths and some number c of closed loops. If d is the k-diagram with the edges that are the paths in this graph but with the closed loops removed, then the product d1d2 is given by d1d2=ηcd. For example, if d1= andd2= , then d1d2= =x2 .

The Brauer algebra Bk(x) is the span of the k-diagrams with multiplication given by the linear extension of the diagram multiplication. The dimension of the Brauer algebra is dim(Bk(x))= (2k)!!= (2k-1)(2k-3) 3·1, since the number of k-diagrams is (2k)!!.

The diagrams in Bk(x) which have all their edges connecting top vertices to bottom vertices form a symmetric group Sk. The elements si= i i+1 andei= i i+1 , 1ik-1, generate the Brauer algebra Bk(x).

Theorem B6.1. The Brauer algebra Bk(x) has a presentation as an algebra by generators s1,s2,,sk-1, e1,e2,,ek-1 and relations si2=1, ei2=xei, eisi=siei =ei, 1ik-1, sisj=sjsi, siej=ejsi, eiej=ejei, |i-j|>1, sisi+1si=si+1sisi+1, eiei+1ei=ei, ei+1eiei+1=ei+1, 1ik-2, siei+1ei=si+1ei, ei+1eisi+1=ei+1si, 1ik-2.

There are two different Brauer algebra analogues of the Schur Weyl duality theorem, Theorem A5.1. In the first one the orthogonal group O(n,) plays the same role that GL(n,) played in the Sk-case, and in the second, the symplectic group Sp(2n,) takes the GL(n,) role.

Let O(n,)={AMn()|AAt=I} be the orthogonal group and let V be the usual n-dimensional representation of the group O(n,). There is an action of the Brauer algebra Bk(n) on Vk which commutes with the action of O(n,) on Vk.

Theorem B6.2.

(a) The action of Bk(n) on Vk generates EndO(n)(Vk).
(b) The action of O(n,) on Vk generates EndBk(n)(Vk).

Let Sp(2n,) be the symplectic group and let V be the usual 2n-dimensional representation of the group Sp(2n,). There is an action of the Brauer algebra Bk(-2n) on Vk which commutes with the action of Sp(2n,) on Vk.

Theorem B6.3.

(a) The action of Bk(-2n) on Vk generates EndSp(2n,)(Vk).
(b) The action of Sp(2n,) on Vk generates EndBk(-2n)(Vk).

Theorem B6.4. The Brauer algebra Bk(x) is semisimple if x{-2k+3,-2k+2,,k-2}.

Partial results for Bk(x)

The following results giving answers to the main questions (Ia-c) for the Brauer algebras hold when x is such that Bk(x) is semisimple.

I. What are the irreducible Bk(x)-modules?

(a) How do we index/count them?
There is a bijection Partitions ofk-2h, h=0,1,,k/2 1-1 Irreducible representationsBλ.
(b) What are their dimensions?
The dimension of the irreducible representation Bλ is given by dim(Bλ) = # of up-down tableaux of shapeλ and lengthk = (k2h) (2h-1)!! (k-2h)! xλhx , where hx is the hook length at the box x in λ. An up-down tableau of shape λ and length k is a sequence (=λ(0),λ(1),λ(k)=λ) of partitions, such that each partition in the sequence differs from the previous one by either adding or removing a box.
(c) What are their characters?
A Murnaghan-Nakayama type rule for the characters of the Brauer algebras was given in [Ram1995].


(1) The Brauer algebra was defined originally by R. Brauer [Bra1937] in 1937. H. Weyl treats it in his book [Wey1946].
(2) The Schur-Weyl duality type theorems are due to Brauer [Bra1937], from his original paper. See also [Ram1995] for a detailed description of these Brauer algebra actions.
(3) The theorem giving values of x for which the Brauer algebra is semisimple is due to Wenzl, see [Wen1988-2].

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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