Conjugacy classes of regular integer matrices

Abstract

This paper is devoted to the theory of $GL_n({\mathbb Z})$-conjugacy classes of regular integer $n\times n$ matrices. Such a matrix is $GL_n({\mathbb Q})$-conjugate to the companion matrix of its characteristic polynomial. But the set of $GL_n({\mathbb Z})$-conjugacy classes of regular integer matrices with a fixed characteristic polynomial $f$ is usually nontrivial (finite if $f$ has simple roots, infinite if $f$ has multiple roots). It is in 1:1-correspondence to a subsemigroup of a certain quotient semigroup of the commutative semigroup of full lattices in the algebra ${\mathbb Q}[t]/(f)$. In its first part, the paper gives a survey on old and new results on full lattices and orders in a finite dimensional commutative ${\mathbb Q}$-algebra with unit element and on induced semigroups. In its longer second part, the paper applies this theory to many examples, essentially all cases with $n=2$, many cases with $n=3$ and two cases with arbitrary $n$, the case with $n$ different integer eigenvalues and the case of a single $n\times n$ Jordan block.