The skew-growth function on the monoid of square matrices

Abstract

We study an elementary divisibility theory for the monoid M(n,R)×:={X∈M(n,R)|det(X)≠0}, where R is a principal ideal domain and M(n,R) is the ring of n-by-n matrices with coefficients in R. We prove that any finite subset ofM(n,R)×has the right least common multiple up to a left unit factor.As an application, we consider the signed generating series, denoted by NM(n,R)×,deg(t) and called the skew-growth function, of least common multiples of all finite sets of irreducible elements of M(n,R)×, assuming R is residue finite. Then, using the above divisibility theory, we show the Euler product decomposition of the skew-growth function:NM(n,R)×,deg(exp(−s))=∏p∈{primes of R}(1−N(p)−s)(1−N(p)−s+1)⋯(1−N(p)−s+n−1) Here N(p):=#(R/(p)) is the absolute norm of p∈R (there is an unfortunate coincidence of notation “N” for the absolute norm and for the skew growth function [6]).