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 Sch├╝tzenberger 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.


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