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arXiv · 2511.02649

A geometric and generating function approach to plethysm

Álvaro Gutiérrez·Rosa Orellana·Franco Saliola·Anne Schilling·Mike Zabrocki
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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.

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Álvaro Gutiérrez, Rosa Orellana, Franco Saliola, Anne Schilling, Mike Zabrocki. 2025-11-04. A geometric and generating function approach to plethysm. https://arxiv.org/abs/2511.02649

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