## Matrices connected with Brauer’s centralizer algebras.(English)Zbl 0849.20009

Electron. J. Comb. 2, Research paper R23, 40 p. (1995); printed version J. Comb. 2, 345-384 (1995).
Summary: In a 1989 paper, Hanlon and Wales showed that the algebra structure of the Brauer centralizer algebra $$A_f^{(x)}$$ is completely determined by the ranks of certain combinatorially defined square matrices $$Z^{\lambda/\mu}$$, whose entries are polynomials in the parameter $$x$$. We consider a set of matrices $$M^{\lambda/\mu}$$ found by Jockusch that have a similar combinatorial description. These new matrices can be obtained from the original matrices by extracting the terms that are of “highest degree” in a certain sense. Furthermore, the $$M^{\lambda/\mu}$$ have analogues $${\mathcal M}^{\lambda/\mu}$$ that play the same role that the $$Z^{\lambda/\mu}$$ play in $$A_f^{(x)}$$, for another algebra that arises naturally in this context.
We find very simple formulas for the determinants of the matrices $$M^{\lambda/\mu}$$ and $${\mathcal M}^{\lambda/\mu}$$, which prove Jockusch’s original conjecture that $$\text{det }M^{\lambda/\mu}$$ has only integer roots. We define a Jeu de Taquin algorithm for standard matchings, and compare this algorithm to the usual Jeu de Taquin algorithm defined by Schützenberger for standard tableaux. The formulas for the determinants of $$M^{\lambda/\mu}$$ and $${\mathcal M}^{\lambda/\mu}$$ have elegant statements in terms of this new Jeu de Taquin algorithm.

### MSC:

 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory 15A15 Determinants, permanents, traces, other special matrix functions
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