On Smith normal forms of $q$-Varchenko matrices

TL;DR

This paper proves that certain symmetric hyperplane arrangements' $q$-Varchenko matrices have a Smith normal form over $ Z[q]$, including arrangements related to regular polygons, dihedral groups, and Platonic solids.

Contribution

It establishes the existence of Smith normal forms over $ Z[q]$ for $q$-Varchenko matrices of symmetric hyperplane arrangements, extending previous results.

Findings

Smith normal form exists over $ Z[q]$ for these matrices

Explicit congruent transformation matrices are constructed

Results apply to arrangements of regular polygons and Platonic solids

Abstract

In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the hyperplane arrangement for the regular $n$-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the $q$-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over $\mathbb Z[q]$ and realize their congruent transformation matrices over $\mathbb Z[q]$ as well.