Diagram algebras, dominance triangularity, and skew cell modules

TL;DR

This paper develops an abstract axiomatic framework for diagram algebras, establishing dominance triangularity of basis transitions and constructing skew cell modules to facilitate tableau-based approaches to the Kronecker problem.

Contribution

It introduces a novel axiomatic framework for diagram algebras, including bases and modules, advancing the study of the Kronecker problem through a tableaux theoretic perspective.

Findings

Transition matrix between bases is dominance unitriangular

Constructed analogues of skew Specht modules

Proposed a tableau framework for the Kronecker problem

Abstract

We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.