On complete functions in Jucys-Murphy elements

TL;DR

This paper provides a combinatorial proof for algebraic formulas related to symmetric functions evaluated at Jucys-Murphy elements, introduces new formulas, and extends results to Hecke algebras, solving a conjecture on orthogonal Weingarten functions.

Contribution

It offers a combinatorial proof of Lassalle's induction relations, derives new simpler formulas, and extends these results to Hecke algebras, addressing a conjecture in the process.

Findings

Combinatorial proof of Lassalle's induction relations

New formulas for class expansion in symmetric functions

Solution to a conjecture on orthogonal Weingarten functions

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, M. 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 $(S_{2n},H_n)$ and use it to solve a conjecture of S. Matsumoto on the subleading term of orthogonal Weingarten function. Finally, we propose a conjecture for a continuous interpolation between both problems.