The partition algebra and the plethysm coefficients I: stability and Foulkes' conjecture

TL;DR

This paper introduces a novel approach using the partition algebra and Schur-Weyl duality to explain the stability of plethysm and Kronecker coefficients, and proves a strengthened Foulkes' conjecture for stable plethysm coefficients.

Contribution

It presents a new method connecting partition algebra and Schur-Weyl duality to analyze plethysm coefficients and proves a strengthened Foulkes' conjecture in a simple, elementary way.

Findings

Explains stability properties of plethysm and Kronecker coefficients.

Proves the strengthened Foulkes' conjecture for stable plethysm coefficients.

Introduces a new algebraic approach to study plethysm.

Abstract

We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This allows us to explain the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion.