Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function

Details Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function

2008-11-21

arxiv.org/abs/0811.3595v2

Mathematics

Combinatorics

12 Pages, no figures

Scientific paper

A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that $h_r(J_1,...,J_n),$ the complete homogeneous symmetric polynomial of degree $r$ in the JM elements, coincides with the $r$th term in the asymptotic expansion of the Weingarten function. We use this connection to determine precisely which conjugacy classes occur in the class basis resolution of $h_r(J_1,...,J_n),$ and to explicitly determine the coefficients of the classes of minimal height when $r < n.$ These coefficients, which turn out to be products of Catalan numbers, are governed by the Moebius function of the non-crossing partition lattice $NC(n).$

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