### Matrices connected with Brauer's centralizer algebras

#### Abstract

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 ${\cal M}^{\lambda / \mu}$ 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 ${\cal M}^{\lambda / \mu}$, which prove Jockusch's original conjecture that $\det M^{\lambda / \mu}$ 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 ${\cal M}^{\lambda / \mu}$ have elegant statements in terms of this new Jeu de Taquin algorithm.