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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.

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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