Combinatorial and algebraic structure in Orlik-Solomon algebras

Abstract

The Orlik-Solomon algebra ${\cal A}(G)$ of a matroid $G$ is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing $G$. On the other hand, some features of the matroid $G$ are reflected in the algebraic structure of ${\cal A}(G)$. In this mostly expository article, we describe recent developments in the construction of algebraic invariants of ${\cal A}(G)$. We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated.