Complete Symmetric Polynomials in Jucys-murphy Elements and the Weingarten Function

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 hr(J1,...,Jn), the complete homogeneous symmetric polynomial of degree r in the JM elements, coincides with the rth 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 hr(J1,...,Jn), 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).