Author:
Boulware Naomi, ,Jing Naihuan,Misra Kailash C., ,
Abstract
In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.
Publisher
Luhansk Taras Shevchenko National University
Subject
Discrete Mathematics and Combinatorics,Algebra and Number Theory
Reference14 articles.
1. [1] N. G. Boulware, Hyperplane arrangements and q-Varchenko matrices, Ph.D. thesis, N. C. State University, 2018.
2. [2]T. W. Cai, Y. Chen, L. Mu, On the Smith normal form of the q-Varchenko matrix of a real hyperplane arrangement, Ars Math. Contemp. 19 (2020), 351-362.
3. [3]G. Denham and P. Hanlon, On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement, Pac. J. Math. 181 (1997), 123-146.
4. [4]Y. Gao and Y. Y. Zhang, Diagonal form of the Varchenko matrices, J. Alg. Combin. 48 (2018), 1-18.
5. [5]B. Grünbaum, Arrangements of hyperplanes, in: Proc. Second Louisiana Conf. on Combinatorics and Graph Theory, Baton Rouge, pp. 41-106, 1971.