Plethysm and lattice point counting

TL;DR

This paper uses lattice point counting techniques to analyze plethysm multiplicities in $GL(n)$, providing explicit formulas, asymptotic insights, and proving a longstanding conjecture by Howe.

Contribution

It introduces a novel lattice point counting approach to plethysm, enabling explicit formulas and asymptotic analysis, and proves Howe's conjecture on the leading term.

Findings

Proved Howe's conjecture on the leading term of plethysm.

Derived explicit formulas for partitions of 3, 4, and 5.

Provided asymptotic growth insights for plethysm multiplicities.

Abstract

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $\mu$ of 3,4, or 5 we obtain an explicit formula in $\lambda$ and $k$ for the multiplicity of $S^\lambda$ in $S^\mu(S^k)$.