Matrices connected with Brauer's centralizer algebras

In a 1989 paper [HW1], Hanlon and Wales showed that the algebra structure of the Brauer Centralizer Algebra A f is completely determined by the ranks of certain combinatorially defined square matrices Z, whose entries are polynomials in the parameter x. We consider a set of matrices M 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 have analogues M that play the same role that the Z play in A f , for another algebra that arises naturally in this context. We find very simple formulas for the determinants of the matrices M andM, which prove Jockusch’s original conjecture that detM 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 andM have elegant statements in terms of this new Jeu de Taquin algorithm.