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At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a certain irreducible summand" of the Birman-Murakami-Wenzl (BMW) representations. It is also mentioned that the explicit correspondence was worked out by a paper of Zinno.

My question: is anyone aware if a computation exists for the matrices of any other irreducible summands of the BMW representations. In other words, I would actually like to see the explicit matrix for some summands of the BMW representation.


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1 Answer 1

up vote 6 down vote accepted

I have not seen this written down but this can certainly be done explicitly. The construction assumes you have explicit matrices for the representations of the symmetric group. Here you have a choice between Young's orthogonal form (which has rational entries) and Specht modules (which have integer entries).

Once you have sorted this out you can construct explicit matrices for the representations of BMW using a Morita equivalence approach. The irreducible representations of the $r$ string algebra are indexed by partitions $\lambda$ with $r-|\lambda|\ge 0$ and even. Let $V(\lambda)$ be the representation of the $|\lambda|$-string symmetric group. let $M$ be the vector space of Brauer diagrams with $r$ points on the top, $|\lambda|$ points on the bottom, with precisely $\lambda$ through strings and such that no two through strings intersect. Then the $r$-string BMW algebra acts on $V(\lambda)\otimes M$.

More detail:

Here is the theory behind the construction. First I realised I am thinking of the Brauer category whereas OP asks about BMW. This basically means the symmetric group algebra should be replaced by the Hecke algebra.

The Brauer category has an increasing sequence of ideals. The ideal $I(k)$ is spanned by diagrams with at most $k$ through strings. The quotient $I(k)/I(k-1)$ is spanned by diagrams with precisely $k$ through strings. Then the key idea is that the category $I(k)/I(k-1)$ is Morita equivalent to the group algebra of the $k$-string symmetric group. The functor from representations of the $k$-string symmetric group to representations of the $k+2c$ string Brauer algebra has a straightforward explicit description.

Even more detail:

The Morita equivalence I have in mind is an extension of the usual equivalence. Let $A$ be a finite dimensional unital algebra and $e\in A$ an idempotent. Let $J$ be the ideal $AeA$ and $B$ the unital algebra $eAe$. Then $J$ and $B$ are Morita equivalent since we have the bimodules $eA$ and $Ae$. Here $J$ is not unital but does have the weaker property that $JJ=J$. Then a right $J$-module $M$ is required to satisfy $MJ=M$. Something like this but without mentioning Morita theory is in Sandy Green's SLN text on Schur algebras.

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I don't quite follow this. In your $V(\lambda)\otimes M$, is the tensor product over the ground field? If so, I don't see how the BMW category acts. For example, take $r = |\lambda| = 2$. –  Kevin Walker Jun 26 '12 at 13:04
Hi @Bruce - I must admit that I am not familiar with Morita equivalence beyond the definition. Are you aware of the reference for the above construction so I can see it in profuse details. Thank you very much!! –  Elden Elmanto Jun 26 '12 at 20:37
Now that you have clarified that you were thinking Brauer, not BMW, in the original post, it makes good sense to me. Thanks for clearing things up. –  Kevin Walker Jun 27 '12 at 0:28
@Elden As far as I know this has not been written up. –  Bruce Westbury Jun 27 '12 at 2:16
@Bruce. Ah that is unfortunate - I will try to read enough to make sense of what you wrote. I suppose I should accept the answer anyway since no one seems to be aware of such a computation and neither does google. Once again, thanks! –  Elden Elmanto Jun 27 '12 at 2:23

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