On Smith normal forms of \(q\)-Varchenko matrices

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.