On the weight space representations of the Brauer algebras

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

Last update: 10 September 2013

The following notes are not intended to be complete in any sense. They are merely facts that I don't wish to forget.

Weight space representations

Let n be a positive integer and let I={-n,-(n-1),,-2,-1,0,1,2,,n-1,n}. Let {vi|iI} be a set of independent noncommuting variables. Define V to be the vector space over with basis {vi|iI}, and define Vm=-span { vi1 vi2 vim| ikI } , so that the words (simple tensors) vi1vi2vim are a basis of Vm.

Let x1,x2,,xn be commuting, independent variables. Define x0=1 and x-i=xi-1 for i=1,2,,n, so that xi is defined for each iI. Define the weight of each word vi1vim of Vm to be wt(vi1vim) =xi1xim. Note that the weight of a word is always of the form xa=x1a1x2a2xnan where a=(a1,a2,,an)n. For each sequence an define (Vm)a= -span { vi1vim |wt (vi1vim) =xa } .

For 0km-1, define an action of the generators Gk and Ek of Bm(2n+1) on Vm by (vi1vi2vim) Gk = vi1vik-1 vik+1vik vik+2 vim, (vi1vi2vim) Ek = δik,-ik+1 jIvi1 vik-1vj v-jvik+2 vim. (1.1) By writing out explicitly the action of a general m-diagram one checks easily that the action defined in (1.1) extends to a well-defined action of Bm(2n+1) on Vm. Since the action of the Brauer algebra on Vm does not change weights of the the words, (Vm)a is always a Bm(n) submodule of Vm.

Let Hn denote the hyperoctahedral group of n×n signed permutation matrices given in the usual way by generators and relations. Define an action of Hn on the variables vi,iI by sivj= { v±(i+1), ifj=±i, v±i, ifj=±(i+1), vj, otherwise, for1in-1, andsnvj= { vn, ifj=±n, vj, otherwise. and define an action of Wn on Vm by w(vi1vim)=vw(i1)vw(im). Define an action of Hn on monomials xi1xim and on sequences a=(a1,,an)n by requiring that for all words vi1vim and wHn, Ifwt (vi1vim)= x1a1xnan= xa,thenwt (w(vi1vim)) =w(xa)=xwa. (1.2)

For each an define λ(a) to be the partition determined by rearranging the sequence (|a1|,|a2|,,|an|) into decreasing order. Then (Vm)a (Vm)λ(a), as Bm(2n+1) modules.


This is clear one only needs to check that the action of Hn and of Bm(2n+1) commute and that the action of Bm(2n+1) preserves weights.

Let λ=(λ1,,λr), λ1λr>0 be a partition of m. We say that the subalgebra Bλ(2n+1)= Bλ1(2n+1) Bλr(2n+1) is a "Young subalgebra" of the Brauer algebra Bm(2n+1). Given a representation M of the Young subalgebra Bλ(2n+1) the induced representation of Bm(2n+1) is given by Bm(2n+1)Bλ(2n+1)M.

In general the weight space representation (Vm)μ is not isomorphic to an induced representation from a Young subalgebra of Bm(2n+1). In fact if μ is a partition of m then (Vm)μ is never an induced representation from a Young subalgebra.


Let us use the notation of symmetric functions. Let sbλ(x1±1,,xn±1,1) denote the orthogonal Schur function which describes the character of the irreducible SO(2n+1) module indexed by λ and let mbλ=γHnλxγ denote the monomial symmetric function corresponding to the partition λ. Here Hn is the hyperoctahedral group. Let the weight multiplicities in the irreducibles are given by coefficients Kλμ such that sbλ=μ Kλμmbμ. It follows from the fact that Kλμ=0 unless μλ in dominance (for the root system of type B) that Kλμ=0if |λ|<|μ|. (1.5)

Let βμ denote the character of the Brauer algebra Bm(2n+1) action on the weight space (Vm)μ. Then it is easy to see from the Schur-Weyl duality that βμ=λBˆm χλKλμ where Bˆm is an index set for the irreducibles of Bm(2n+1) and χλ is the irreducible character of the Brauer algebra Bm(2n+1). If μ is a partition of m then (1.5) βμ= λm χλKλμ (1.6)

Now let us use the Frobenius characteristic map. Since induction from Young subalgebras of the Brauer algebra corresponds to taking tensor products of SO(2n+1) representations under the Frobenius characteristic map, the βμ will be an induced representation from a Young subalgebra Bλ(2n+1), λ=(λ1,,λr) if and only if there are symmetric functions H1,H2,,Hr such that

(1.7a) Hj=νBˆλjcjνsbν for some nonnegative integers cjν, and
1.7b) H1H2Hr= λBˆmsbλKλμ.
Furthermore, if we also require that μm then (1.5) forces
1.7c) H1H2Hr= λmsbλKλμ.
Now we note three facts: If νs and πp then the decomposition of sbνsbπ
1.8a) Contains a nonzero term sbγ for some γs+t
(1.8b) Contains a nonzero term sbγ for some partition γs+t-2.
1.8c) Does not contain a nonzero term sbγ for any partition |γ|>s+t.
Both of these facts can be proved easily by looking at the decomposition rule of Black-King-Wybourne [BKW1983] given in Theorem 5.3 of [Sun1990].

Let μm and assume that symmetric functions H1,,Hr exist satisfying (1.7abc). Then (1.8c) combined with (1.7c) implies that Hj=νBˆλjcjνsbν where cjν>0 for at least one νλj. Then (1.8b) implies that H1Hr contains a nonzero term sbγ for some partition γs+t-2. This is a contradiction to (1.7c). Thus βμ is not equal to a character induced from a Young subalgebra.

For each partition λBˆm, the dimension of the λ weight space is dim((Vm)λ)= s0+2(s1++sn)=m-|λ| ( m s0,λ1+s1, s1,λ2+s2, s2,,λ+sn ,sn )


A basis vector w=vi1vim in Vm is of weight λ=(λ1,,λn) if there is a sequence of positive integers s0,s1,,sn such that w contains s0 v0's, λ1+s1 v1's, s1 v-1's, and so on. The multinomial coefficient is just the number of ways of choosing the positions of these letters.

In type C the dimension of V is 2n and there are no vectors v0 of weight 0 in V. The dimension of the zero weight space in type C is given by dim((Vm)0)= { 0, ifmis odd, (2r)!r!2 s1++sn=r (rs1,s2,,sn)2 , ifm=2r.


This is clear from (1.9) and the fact that (2rs1,s2,,sn) = (2r)! s1!s1! sn!sn! =(2r)!r!2 (rs1,s2,,sn)2 .

Notes and References

This is a copy of the paper On the weight space representations of the Brauer algebras by Arun Ram, Department of Mathematics, University of Wisconsin, Madison, WI 53706, January 20, 1994. This paper was supported in part by a National Science Foundation postdoctoral fellowship.

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