The Varchenko Determinant of a Coxeter Arrangement

TL;DR

This paper derives an explicit formula for the Varchenko determinant of hyperplane arrangements associated with finite Coxeter groups, enabling more efficient computation of these determinants.

Contribution

It provides the first explicit formula for the Varchenko determinant in the context of finite Coxeter group arrangements, improving computational methods.

Findings

Explicit formula for the Varchenko determinant of Coxeter arrangements

Identification of properties of hyperplane intersections with chambers

Enhanced understanding of determinant factorization in hyperplane arrangements

Abstract

The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.