On Complete Functions in Jucys-Murphy Elements

Abstract

The problem of computing the class expansion of some symmetric functions evaluated in Jucys-Murphy elements appears in different contexts, for instance, in the computation of matrix integrals. Recently, Lassalle gave a unified algebraic method to obtain some induction relations on the coefficients in this kind of expansion. In this paper, we give a simple purely combinatorial proof of his result. Besides, using the same type of argument, we obtain new simpler formulas. We also prove an analogous formula in the Hecke algebra of (S2n, Hn) and use it to solve a conjecture of Matsumoto on the subleading term of orthogonal Weingarten function. Finally, we propose a conjecture for a continuous interpolation between both problems.