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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.27982 |
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| _version_ | 1866911723591565312 |
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| author | Steinmann, Isabelle |
| author_facet | Steinmann, Isabelle |
| contents | The goal of this paper is to understand the set $\mathrm{End}(W)$ of endomorphisms of an irreducible spherical reflection group $W$. We do this in two ways: numerically, by deriving an explicit formula for $|\mathrm{End}(W)|$; and probabilistically, by exploring the question \textit{what does a random endomorphism of $W$ look like?} For example, we show that as $n\to\infty$ the probability that a random endomorphism of $W_n$ is an automorphism tends to $\frac{1}{2}$ if $W_n=C_{2n}$ or $D_n$, to $\frac{1}{4}$ if $W_n=C_{2n+1}$, and to $1$ if $W_n=A_n.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27982 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random endomorphisms of spherical reflection groups Steinmann, Isabelle Group Theory The goal of this paper is to understand the set $\mathrm{End}(W)$ of endomorphisms of an irreducible spherical reflection group $W$. We do this in two ways: numerically, by deriving an explicit formula for $|\mathrm{End}(W)|$; and probabilistically, by exploring the question \textit{what does a random endomorphism of $W$ look like?} For example, we show that as $n\to\infty$ the probability that a random endomorphism of $W_n$ is an automorphism tends to $\frac{1}{2}$ if $W_n=C_{2n}$ or $D_n$, to $\frac{1}{4}$ if $W_n=C_{2n+1}$, and to $1$ if $W_n=A_n.$ |
| title | Random endomorphisms of spherical reflection groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.27982 |