A geometric and generating function approach to plethysm

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 ${\mathrm{S}\mathrm{L}}_2$-plethysm coefficients.