A geometric and generating function approach to plethysm

Abstract

Plethysm coefficients ${\mathsf{a}}_{\hat{I}¼[\hat{I}½]}^{\hat{I}}»$ are the structure coefficients of the plethysm of Schur functions $s_{\hat{I}}¼[s_{\hat{I}}½]={\sum }_{\hat{I}}»{\mathsf{a}}_{\hat{I}¼[\hat{I}½]}^{\hat{I}}»s_{\hat{I}}»$. We study a bivariate generating function of plethysm coefficients when $\hat{I}»$ 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.

33 pages; v2: added more results to Section 6 and Appendix B