Matroidal Schur algebras

Motivated by a geometric description of the Schur algebra due to the second author, we define for any matroid M and principal ideal domain k , a quasi-hereditary algebra R ( M ) defined over k which we call a matroidal Schur algebra. We show that the Ringel dual of R ( M ) is the matroidal Schur algebra R(M^*) associated with the dual matroid M^* . When k is a field of characteristic zero, the algebra R ( M ) can be seen as a categorification of a formula due to Kook–Reiner–Stanton and related work of Denham. More generally, the behavior of R ( M ) in positive characteristic is closely related to determinant formulas of Schechtman–Varchenko and Brylawski–Varchenko.