On the Smith normal form of Varchenko matrix

Abstract

Let A be a hyperplane arrangement in A isomorphic to R n. Let V q be the q-Varchenko matrix for the arrangement A with all hyperplane parameters equal to q. In this paper, we consider three interesting cases of q-Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over Z[q].

On Smith normal forms of q-Varchenko matrices

“…. , h n }, was presented in [2]. Here the hyperplanes consists of all facets of the regular n-polygons.…”

“…Explicitly, we will study the related hyperplane arrangement models based on Platonic polyhedra and their degenerations. The cyclic model has been studied in a recent paper [2] as an example of peelable hyperplane arrangement. In this paper we will use a different method to approach all hyperplane arrangement models based on regular polyhedra, and show that all of the q-Varchenko matrices have the Smith normal forms over Z[q] and also the congruent transformations can be realized in Z[q] as well.…”