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

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 ${\Gamma }_{(t|X)}{s}_{\alpha }=\sigma [tX]s_{\alpha }[x-1/t]$ 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 $n$.