Category $\mathcal{O}$ for Oriented Matroids

TL;DR

This paper constructs a new class of finite-dimensional algebras from oriented matroid programs, generalizing known algebraic structures related to hyperplane arrangements and exploring their properties in both Euclidean and non-Euclidean cases.

Contribution

It introduces a novel algebraic framework associated with oriented matroid programs, extending existing theories to nonlinear and non-Euclidean settings.

Findings

Algebras are quasi-hereditary and Koszul in Euclidean cases.

Non-Euclidean algebras are not quasi-hereditary or Koszul but have standard modules.

The construction recovers known algebras from hyperplane arrangements.

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 nonlinear 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.