Simple modules for the partition algebra and monotone convergence of Kronecker coefficients

TL;DR

This paper constructs bases for simple modules of partition algebras using alcove geometry, providing a new proof of the monotone convergence of Kronecker coefficients through stratifications of cell modules.

Contribution

It introduces a geometric approach to understanding simple modules of partition algebras and offers a novel proof of Kronecker coefficient convergence.

Findings

Bases for simple modules indexed by alcove paths

New proof of monotone convergence of Kronecker coefficients

Stratification of cell modules elucidates module structure

Abstract

We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker coefficients using stratifications of the cell modules of the partition algebra.