A comment on 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 $n$.