A category $\mathcal{O}$ for oriented matroids

Abstract

We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category $\mathcal{O}$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster.

Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.