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RIMS-1755 Monoid structure on square matrices over a PID By

By  and Kyoji Saito and Kyoji Saito

Abstract

We consider the set M(n, R) × of all square matrices of size n ∈ Z≥1 with non-zero determinants and coefficients in a principal ideal domain R. It forms a cancellative monoid with the matrix product. We develop an elementary theory of divisions by irreducible elements in M(n, R) × , and show that any finite set of irreducible elements of M(n, R) × has the right/left least common multiple up to a unit factor. As an application, we calculate the growth function P M(n,R) ×,deg(t) and the skew growth function N M(n,R) ×,deg(t) of the monoid M(n, R) ×. We get expressions P M(n,R) ×,deg(exp(−s)) = ζR(s)ζR(s−1) · · ·ζR(s−n+1) and N M(n,R) ×,deg(exp(−s)) = p∈{primes} (1−N(p)s)(1−N(p) s−1) · · · (1−N(p) s−n+1), where ζR(s) is Dedekind zetafunction and N is the absolute norm on R. The structure of least common multiples in the monoid M(n, R) × studied above gives an elementary and direct proof of these decompositions, that is distinct from proofs by classical machinary. Content

Year: 2012
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