The number of irreducibles in the plethysm $s_λ[s_m]$
Ming Yean Lim
TL;DR
The paper develops a representation-theoretic and polyhedral framework to count irreducibles in the plethysm $s_\lambda[s_m]$ by summing plethysm coefficients over $\nu$, connecting this sum to a character $N^m$ of $\mathfrak{S}_n$ and to Ehrhart theory via lattice-point counts in polytopes. It proves that $N^m$ is a genuine character and that $\langle \chi^\lambda, N^m \rangle = \sum_{\nu} a_{\lambda,m}^\nu$, with $N^m(\sigma)$ an Ehrhart quasipolynomial in $m$; for $\lambda=n$ there is a combinatorial interpretation in terms of permutation-equivalence classes of certain matrices and a relation to Foulkes' conjecture. The authors develop a general $(G \wr C_2)$-set and representation framework, derive a twisted orbit-counting formula, and decompose $N^m$ into contributions from classes in $T(n,m)/\sim$, enabling asymptotic and special-case analyses (notably $m=2$). The work yields new avenues for interpreting plethysm coefficients combinatorially and connects to polyhedral geometry through Ehrhart theory, with implications for Foulkes-type inequalities and asymptotics of plethysm sums.
Abstract
We give a formula for the number of irreducibles (with multiplicity) in the decomposition of the plethysm $s_λ[s_m]$ of Schur functions in terms of the number of lattice points in certain rational polytopes. In the case where $λ= n$ consists of a single part, we will give a combinatorial interpretation of this number as the cardinality of a set of matrices modulo permutation equivalence. This is also the setting of Foulkes' conjecture, and our results allow us to state a weaker version that only involves comparing the cardinalities of such sets, rather than the multiplicities of irreducible representations.