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Conjugacy classes of regular integer matrices

Claus Hertling, Khadija Larabi

TL;DR

This work investigates GL$_n$(Z)-conjugacy classes of regular integer matrices, linking them to full lattices and orders in finite-dimensional Q-algebras via the cyclic algebra A$_f$=Q[t]/(f). It shows that for fixed f the class set S$_{1,f}$ is finite if f has distinct roots and infinite otherwise, with a 1:1 correspondence to ε-classes of full lattices in A$_f$ and an intrinsic semigroup structure. The first half surveys semigroup and lattice-order theory, while the second half applies this framework to a broad array of examples, notably all n=2 irreducible cases, many n=3 cases, and select n≥3 cases with one Jordan block or 1-dimensional summands. The results unify number-theoretic and lattice-theoretic methods (including binary quadratic forms and Conway’s topograph) to classify conjugacy classes and their finer invariants, providing explicit normal forms, order computations, and multiplication/division tables in several low-dimensional instances. Overall, the paper extends Jordan-Zassenhaus-type classifications to nonsemisimple regular integer matrices, clarifying when finiteness holds and how to parametrize infinite families via lattice-theoretic data, with potential applications to arithmetic groups and representation theory.

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.

Conjugacy classes of regular integer matrices

TL;DR

This work investigates GL(Z)-conjugacy classes of regular integer matrices, linking them to full lattices and orders in finite-dimensional Q-algebras via the cyclic algebra A=Q[t]/(f). It shows that for fixed f the class set S is finite if f has distinct roots and infinite otherwise, with a 1:1 correspondence to ε-classes of full lattices in A and an intrinsic semigroup structure. The first half surveys semigroup and lattice-order theory, while the second half applies this framework to a broad array of examples, notably all n=2 irreducible cases, many n=3 cases, and select n≥3 cases with one Jordan block or 1-dimensional summands. The results unify number-theoretic and lattice-theoretic methods (including binary quadratic forms and Conway’s topograph) to classify conjugacy classes and their finer invariants, providing explicit normal forms, order computations, and multiplication/division tables in several low-dimensional instances. Overall, the paper extends Jordan-Zassenhaus-type classifications to nonsemisimple regular integer matrices, clarifying when finiteness holds and how to parametrize infinite families via lattice-theoretic data, with potential applications to arithmetic groups and representation theory.

Abstract

This paper is devoted to the theory of -conjugacy classes of regular integer matrices. Such a matrix is -conjugate to the companion matrix of its characteristic polynomial. But the set of -conjugacy classes of regular integer matrices with a fixed characteristic polynomial is usually nontrivial (finite if has simple roots, infinite if 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 . In its first part, the paper gives a survey on old and new results on full lattices and orders in a finite dimensional commutative -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 , many cases with and two cases with arbitrary , the case with different integer eigenvalues and the case of a single Jordan block.
Paper Structure (23 sections, 50 theorems, 341 equations, 12 figures, 11 tables)

This paper contains 23 sections, 50 theorems, 341 equations, 12 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Commutative semigroups
  3. Finite dimensional ${\mathbb Q}$-algebras and multiplicative metrics
  4. Orders, full lattices and their semigroups
  5. Classes of full lattices, two quotient semigroups
  6. Conjugacy classes of matrices and $\varepsilon$-classes of full lattices
  7. The irreducible rank 2 cases
  8. Preliminaries, a correspondence with binary quadratic forms
  9. Algebras, orders, and $SL_2({\mathbb Z})$-conjugacy classes
  10. Normal forms and semi-normal forms
  11. Conway's topograph
  12. An irreducible rank 3 case
  13. The separable cases with 1-dimensional summands
  14. Observations on the case of arbitrary rank
  15. The rank 2 case
  16. ...and 8 more sections

Key Result

Theorem 2.2

DTZ62HL26 Let $S$ be a commutative semigroup. (a) Let $a\in S$ be invertible. Then there is a unique element $c\in S$ with the properties $(ac=a,\exists\ b\in S \textup{ with }ab=c)$. It is called $e_a$. It is idempotent. There is a unique element $b\in S$ with the properties $(ab=e_a,be_a=b)$. It i is a group. It is a maximal subgroup of $S$. Any maximal subgroup of $S$ is equal to $G(\widetilde{

Figures (12)

  • Figure 7.1: Action of $\sigma$ and $\tau$ on $R_2$
  • Figure 7.2: Action of $PSL_2({\mathbb Z})$ on $L_2$ and the maps $\psi_1:R_1\to L_1$ and $\psi_2:R_2\to L_2$
  • Figure 7.3: Encoding the value atlas by labeling regions and oriented edges
  • Figure 7.4: From the values $(\omega_1(\rho_1)$,$\omega_2((\rho_1,\rho_2))$, $\omega_1(\rho_2))$ to the values $\omega_1(\rho_3)$ and $\omega_1(\rho_4)$
  • Figure 7.5: From the values of three neighboring regions to the values of three oriented edges
  • ...and 7 more figures

Theorems & Definitions (63)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Definition 3.3
  • Definition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 53 more