Involutions and the Gelfand character

TL;DR

This paper derives a recursive generating function for the Gelfand character of symmetric groups by analyzing descents in lambda-unimodal involutions, building on a result by Adin, Postnikov, and Roichman.

Contribution

It introduces a new recursive formula for the Gelfand character using combinatorial analysis of involutions and descents.

Findings

Derived a recursive generating function for the Gelfand character.

Connected involution descent analysis to representation theory.

Extended combinatorial methods to symmetric group characters.

Abstract

The Gelfand representation of $\mathcal{S}_n$ is the multiplicity-free direct sum of the irreducible representations of $\mathcal{S}_n$. In this paper, we use a result of Adin, Postnikov, and Roichman to find a recursive generating function for the Gelfand character. In order to find this generating function, we investigate descents of so-called $\lambda$-unimodal involutions.