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 (x) 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 (x) f , for another algebra that arises naturally in this context. We find very simple formulas for the determinants of the matrices M =¯ and M =¯ , which prove Jockusch's original conjecture that det M =¯ 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..