Jucys-Murphy Elements and a Combinatorial Proof of an Identity of S. Kerov

TL;DR

This paper provides a combinatorial proof of the diagonal action of Jucys-Murphy elements in the symmetric group algebra, extending known results and presenting new related findings.

Contribution

It offers a combinatorial proof of Jucys-Murphy elements' properties and introduces additional results derived from these methods.

Findings

Confirmed diagonal action of R_j elements for j=n

Derived new combinatorial identities related to symmetric groups

Extended understanding of Jucys-Murphy elements' properties

Abstract

Consider the elements of the group algebra CS_{n} given by R_{j}=Sigma_{i=1}^{j-1}(ij), for 2<=j<=n. Jucys [3 - 5] and Murphy[7] showed that these elements act diagonally on elements of S_{n} and gave explicit formulas for the diagonal entries. As requested by the late S. Kerov, we give a combinatorial proof of this work in case j=n and present several similar results which arise from these combinatorial methods.