A geometric and generating function approach to plethysm

TL;DR

This paper investigates the structure of plethysm coefficients through generating functions, revealing their rationality, geometric interpretations, and recursive properties, especially for bounded-length partitions.

Contribution

It introduces a rational generating function for plethysm coefficients with bounded length, provides explicit algorithms for length two, and explores geometric and reciprocity properties.

Findings

Generated functions are rational for bounded-length partitions.

Explicit geometric algorithms are provided for length two cases.

Evidence suggests the generating function relates to quantum Ehrhart series.

Abstract

Plethysm coefficients $\mathsf{a}_{\mu[\nu]}^\lambda$ are the structure coefficients of the plethysm of Schur functions $s_\mu[s_\nu] = \sum_{\lambda} \mathsf{a}_{\mu[\nu]}^\lambda s_\lambda$. We study a bivariate generating function of plethysm coefficients when $\lambda$ 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.