Diagonal Form of the Varchenko Matrices

TL;DR

This paper characterizes when the Varchenko matrix of a real hyperplane arrangement can be diagonalized, showing it is possible if and only if the arrangement is semigeneral, and provides an explicit combinatorial method for such cases.

Contribution

It establishes a necessary and sufficient condition for diagonalizability of the Varchenko matrix and offers an explicit combinatorial procedure for the diagonal form in semigeneral arrangements.

Findings

Varchenko matrix has a diagonal form iff the arrangement is semigeneral.

Explicit combinatorial method for diagonalization in semigeneral case.

Provides a combinatorial interpretation of diagonal entries.

Abstract

Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.