A comment of the combinatorics of the vertex operator Γ(t|X)

Description

The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let Γ(t|X) be the vertex operator defined by Γ(t|X)sα=∑n∈Zs(n,α)[X]tn. We provide a combinatorial proof for the identity Γ(t|X)sα=σ[tX]sα[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,α)[X] in the Schur basis, for any integer value of n.

Additional details