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

N. Boulware, N. Jing, K. C. Misra


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.


Smith normal forms, hyperplane arrangements, Platonic solids, Varchenko matrices

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DOI: http://dx.doi.org/10.12958/adm2006


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