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Báo cáo toán học: " Matrices connected with Brauer’s centralizer algebras"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Matrices connected with Brauer’s centralizer algebras. | Matrices connected with Brauer s centralizer algebras Mark D. McKerihan t Department of Mathematics University of Michigan Ann Arbor MI 48109 Submitted October 9 1995 Accepted October 31 1995 Abstract In a 1989 paper HW1 Hanlon and Wales showed that the algebra structure of the Brauer Centralizer Algebra AjX is completely determined by the ranks of certain combinatorially dehned square matrices Zxxu whose entries are polynomials in the parameter x. We consider a set of matrices Mx l 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 Mx r have analogues Mx l that play the same role that the Zx play in Af for another algebra that arises naturally in this context. We hnd very simple formulas for the determinants of the matrices Mx and Mx A which prove Jockusch s original conjecture that det Mxx i has only integer roots. We dehne a Jeu de Taquin algorithm for standard matchings and compare this algorithm to the usual Jeu de Taquin algorithm dehned by Schutzenberger for standard tableaux. The formulas for the determinants of Mxx i and Mx l have elegant statements in terms of this new Jeu de Taquin algorithm. Contents 1 Introduction 2 1.1 Acknowledgments. 8 Subject Class 05E15 05E10 This research was supported in part by a Department of Education graduate fellowship at the University of Michigan 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 2 1995 R23 2 2 Determinants of M and M 8 2.1 Column permutations of standard matchings . 8 2.2 Product formulas for M and M. 12 2.3 Eigenvalues of Tk x and Tk y1 . yn . 13 2.4 The column span of P. 14 2.5 Computation of det M and det M . 19 3 Jeu de Taquin for standard matchings 23 3.1 Definition of the algorithm. 23 3.2 Jeu de Taquin preserves standardness . 25 3.3 Dual Knuth equivalence with JdT for tableaux. 31 3.4 The normal shape obtained via JdT. 35 3.5 An alternate .