The Varchenko Determinant for Oriented Matroids

TL;DR

This paper extends the Varchenko determinant formula from hyperplane arrangements to oriented matroids, showing the determinant depends only on the underlying matroid and providing new topological insights.

Contribution

It generalizes the Varchenko determinant formula to oriented matroids and introduces a topological approach involving supertopes and contractible order complexes.

Findings

Determinant formula for Varchenko matrix extends to oriented matroids.

Determinant depends solely on the underlying matroid.

Supertope order complex is contractible.

Abstract

We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for cones in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest. We show that a supertope considered as a subposet of the tope poset has a contractible order complex.