# 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.

## Key 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 $n$.

---
Source: https://tomesphere.com/paper/1701.02516