Conjugacy classes of regular integer matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911452340682752 |
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| author | Hertling, Claus Larabi, Khadija |
| author_facet | Hertling, Claus Larabi, Khadija |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15748 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conjugacy classes of regular integer matrices Hertling, Claus Larabi, Khadija Rings and Algebras Number Theory Representation Theory 15B36, 15A21, 20M14, 11R54, 16H20 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. |
| title | Conjugacy classes of regular integer matrices |
| topic | Rings and Algebras Number Theory Representation Theory 15B36, 15A21, 20M14, 11R54, 16H20 |
| url | https://arxiv.org/abs/2602.15748 |