A geometric and generating function approach to plethysm

Abstract

Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients.