On commutators of unipotent matrices of index 2
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909320795389952 |
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| author | Rosa, Kennett L. Dela Santos, Juan Paolo C. |
| author_facet | Rosa, Kennett L. Dela Santos, Juan Paolo C. |
| contents | A commutator of unipotent matrices of index 2 is a matrix of the form $XYX^{-1}Y^{-1}$, where $X$ and $Y$ are unipotent matrices of index 2, that is, $X\ne I_n$, $Y\ne I_n$, and $(X-I_n)^2=(Y-I_n)^2=0_n$. If $n>2$ and $\mathbb F$ is a field with $|\mathbb F|\geq 4$, then it is shown that every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on $\mathbb F$ are given that improve the upper bound on the commutator factors from four to three or two. The situation for $n=2$ is also considered. This study reveals a connection between factorability into commutators of unipotent matrices and properties of $\mathbb F$ such as its characteristic or its set of perfect squares. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On commutators of unipotent matrices of index 2 Rosa, Kennett L. Dela Santos, Juan Paolo C. Rings and Algebras 15A21, 15A23, 15B33, 15B99, 20H20 A commutator of unipotent matrices of index 2 is a matrix of the form $XYX^{-1}Y^{-1}$, where $X$ and $Y$ are unipotent matrices of index 2, that is, $X\ne I_n$, $Y\ne I_n$, and $(X-I_n)^2=(Y-I_n)^2=0_n$. If $n>2$ and $\mathbb F$ is a field with $|\mathbb F|\geq 4$, then it is shown that every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on $\mathbb F$ are given that improve the upper bound on the commutator factors from four to three or two. The situation for $n=2$ is also considered. This study reveals a connection between factorability into commutators of unipotent matrices and properties of $\mathbb F$ such as its characteristic or its set of perfect squares. |
| title | On commutators of unipotent matrices of index 2 |
| topic | Rings and Algebras 15A21, 15A23, 15B33, 15B99, 20H20 |
| url | https://arxiv.org/abs/2409.13339 |