Conjugacy classes of regular integer matrices

TL;DR

This paper studies the classification of regular integer matrices under conjugation by integer matrices, linking it to lattice theory and algebraic structures, with detailed results for small dimensions and specific eigenvalue cases.

Contribution

It provides a comprehensive survey of lattice and order theory in algebraic structures and applies this to classify conjugacy classes of integer matrices, especially for small dimensions.

Findings

Finite conjugacy classes for matrices with simple roots

Infinite classes for matrices with multiple roots

Explicit classifications for low-dimensional cases

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.