On the Product of Coninvolutory Affine Transformations
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arXiv
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| Format: | Preprint |
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2026
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| author | Dutta, Sandipan Gongopadhyay, Krishnendu Mondal, Rahul |
| author_facet | Dutta, Sandipan Gongopadhyay, Krishnendu Mondal, Rahul |
| contents | A complex matrix is called \emph{coninvolutory} if $T\overline{T}=I$. In this paper, we study decompositions of affine transformations in $\mathrm{Aff}(n,\mathbb{C})=\mathrm{GL}(n,\mathbb{C})\ltimes \mathbb{C}^n$ into products of coninvolutions. We prove that an affine transformation $g$ is a product of two coninvolutions in $\mathrm{Aff}(n,\mathbb{C})$ if and only if its linear part $L(g)$ is $c$-reversible; that is, $L(g)$ is conjugate to $\overline{L(g)}^{-1}$ in $\mathrm{GL}(n,\mathbb{C})$. Equivalently, $g$ is conjugate to $\overline{g}^{-1}$ in $\mathrm{Aff}(n,\mathbb{C})$. We further characterize elements that are products of three coninvolutions via consimilarity and show that every $g=(A,v)\in \mathrm{Aff}(n,\mathbb{C})$ with $|\det(A)|=1$ can be expressed as a product of at most four coninvolutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Product of Coninvolutory Affine Transformations Dutta, Sandipan Gongopadhyay, Krishnendu Mondal, Rahul Group Theory Operator Algebras Rings and Algebras 15A86 (Primary) 20E45, 51N30 (Secondary) A complex matrix is called \emph{coninvolutory} if $T\overline{T}=I$. In this paper, we study decompositions of affine transformations in $\mathrm{Aff}(n,\mathbb{C})=\mathrm{GL}(n,\mathbb{C})\ltimes \mathbb{C}^n$ into products of coninvolutions. We prove that an affine transformation $g$ is a product of two coninvolutions in $\mathrm{Aff}(n,\mathbb{C})$ if and only if its linear part $L(g)$ is $c$-reversible; that is, $L(g)$ is conjugate to $\overline{L(g)}^{-1}$ in $\mathrm{GL}(n,\mathbb{C})$. Equivalently, $g$ is conjugate to $\overline{g}^{-1}$ in $\mathrm{Aff}(n,\mathbb{C})$. We further characterize elements that are products of three coninvolutions via consimilarity and show that every $g=(A,v)\in \mathrm{Aff}(n,\mathbb{C})$ with $|\det(A)|=1$ can be expressed as a product of at most four coninvolutions. |
| title | On the Product of Coninvolutory Affine Transformations |
| topic | Group Theory Operator Algebras Rings and Algebras 15A86 (Primary) 20E45, 51N30 (Secondary) |
| url | https://arxiv.org/abs/2603.10719 |