Anzahl theorems of matrices over the residue class ring modulo p~s q~t and their applications

摘要

Let h = p(s)q(t) be its decomposition into a product of powers of two distinct primes, and Z(h) be the residue class ring modulo h. In this paper, we determine the Smith normal forms of matrices, compute the number of the orbits of m x n matrices under the action of the group GL(m )(Z(h)) x GL(n )(Z(h)) and the length of each orbit over Z(h). As applications, we determine the clique number, the independence number and the chromatic number of the bilinear forms graph, construct a family of association schemes by using 1 x n matrices and compute the intersection numbers and the character table over Z(h) for s = t = 1. (C) 2019 Elsevier Inc. All rights reserved.