Matroidal Schur Algebras

TL;DR

This paper introduces a new algebraic structure called matroidal Schur algebras associated with matroids, revealing a duality correspondence and connecting to existing work in representation theory and combinatorics.

Contribution

It constructs quasi-hereditary algebras from matroids, establishing a duality correspondence and linking to prior research in representation theory and combinatorics.

Findings

Matroid duality corresponds to Ringel duality in the associated algebras.

The algebras relate to existing work by Schechtman-Varchenko and Brylawski-Varchenko.

In characteristic zero, the algebras connect to Kook-Reiner-Stanton and Denham's work.

Abstract

Fix a principal ideal domain $k$. In this article we associate to a (weighted) matroid $M$ a quasi-hereditary algebra $R(M)$ defined over $k$ such that matroid duality corresponds to Ringel duality of quasi-hereditary algebras. The representation theory of these algebras is related to work of Schechtman-Varchenko and Brylawski-Varchenko. In characteristic zero, our algebras are also closely related to work of Kook-Reiner-Stanton and Denham.