Matroidal Schur Algebras

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.