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The field of definition will be the complex numbers, $V$ is a vector space of dimension $m$, and $O(V)$ is the orthogonal group preserving some nondegenerate bilinear form on $V$. The centralizer algebra of $O(V)$ acting on $V^{\otimes f}$ is a quotient of the Brauer algebra, and there is a decomposition of the form

$V^{\otimes f} = \bigoplus_{\lambda} S_{[\lambda]} V \otimes \pi_\lambda$

where $\lambda$ ranges over all partitions of size $f - 2k$, where $0 \le k \le f/2$, $S_{[\lambda]} V$ is an irrep for the orthogonal group, and $\pi_\lambda$ is an irrep for the centralizer algebra.

I would like to have an explicit construction for $\pi_\lambda$ similar to the Specht modules for the symmetric group, i.e., given by generators and relations, preferably where the generators are given by some combinatorial objects. Is this written down somewhere? The dimension of $\pi_\lambda$ is essentially the number of oscillating tableaux of shape $\lambda$, so a source that touches on that would also be appreciated.

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3 Answers 3

Take the diagram description of the Brauer algebra. This has a sequence of ideals $I(r)$ where $I(r)$ has basis diagrams with at most $r$ through strings. The successive quotients are Morita equivalent to the group algebra of the symmetric group. If you take the Specht module of the symmetric group then the corresponding representation of the Brauer algebra can be described explicitly and meets your requirements.

Before anyone points this out: I know ideals do not have units so the usual theory of Morita equivalence does not apply. Nevertheless this story does make sense.

If this answer is too concise I can give more detail.

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If you take subrepresentation of those tensor whose any contraction with the defining scalar product is zero, then the decomposition looks exactly the same as in $(GL(n),S_n)$ case. One can then decompose $\otimes^k V$ into tracefree tensors and tensors of the form $\oplus g^{ab}U^{\ldots}$ where $U^{\ldots}$ is totally trace-free and $\oplus g^{ab}g^{cd}W^{\ldots}$ where $W^{\ldots}$ are trace-free etc till one gets either the trivial representation corresponding to complete trace of tensor from $\otimes^k V$ or a representation isomorphic to $V$. Each such trace-free part decomposes exactly the same (in the stable range) as $(GL(n),S_n)$-modules. The keyword is harmonic tensors and details, proof and examples can be found in the book by Goodman and Wallach.

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Possibly this is the same thing as Bruce Westbury is saying, but I found this short paper of Kerov, "Realizations of representations of the Brauer semigroup" which constructs the irreducible representations in the semisimple case.

I first thought this was some kind of induced representation, but now I am not so sure. But the construction is still explicit.

I would still like to see something that starts directly with oscillating tableaux, however.

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