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arXiv · 0811.3595

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

Jonathan Novak
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Abstract

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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Jonathan Novak. 2008-11-24. Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function. https://arxiv.org/abs/0811.3595

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