A comment of the combinatorics of the vertex operator $\Gamma_{(t|X)}$

TL;DR

This paper provides a combinatorial proof of a vertex operator identity related to symmetric functions, expanding understanding of the combinatorial structures underlying the Jacobi--Trudi identity and Schur functions.

Contribution

It offers a new combinatorial proof of a vertex operator identity and describes the expansion of certain symmetric functions in the Schur basis for all integer parameters.

Findings

Combinatorial proof of the vertex operator identity

Description of the expansion of $s_{(n,eta)} [X]$ in the Schur basis for all integers n

Overview of combinatorial ideas behind the identity

Abstract

The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let $\Gamma_{(t|X)}$ be the vertex operator defined by $\Gamma_{(t|X)} s_\alpha =\sum_{n \in \mathbb{Z}} s_{(n,\alpha)} [X] t^n$. We provide a combinatorial proof for the identity $\Gamma_{(t|X)} s_\alpha = \sigma[tX] s_{\alpha}\big[x-1/t\big] $ due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of $s_{(n,\alpha)} [X] $ in the Schur basis, for any integer value of $n$.