The relational complexity of linear groups acting on subspaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912144048521216 |
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| author | Freedman, Saul D. Kelsey, Veronica Roney-Dougal, Colva M. |
| author_facet | Freedman, Saul D. Kelsey, Veronica Roney-Dougal, Colva M. |
| contents | The relational complexity of a subgroup $G$ of $\mathrm{Sym}(Ω)$ is a measure of the way in which the orbits of $G$ on $Ω^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between $\mathrm{PSL}_{n}(\mathbb{F})$ and $\mathrm{PGL}_{n}(\mathbb{F})$, for an arbitrary field $\mathbb{F}$, acting on the set of $1$-dimensional subspaces of $\mathbb{F}^n$. We also bound the relational complexity of all groups lying between $\mathrm{PSL}_{n}(q)$ and $\mathrm{P}Γ\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16111 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The relational complexity of linear groups acting on subspaces Freedman, Saul D. Kelsey, Veronica Roney-Dougal, Colva M. Group Theory 20B15, 20G40, 03C13 The relational complexity of a subgroup $G$ of $\mathrm{Sym}(Ω)$ is a measure of the way in which the orbits of $G$ on $Ω^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between $\mathrm{PSL}_{n}(\mathbb{F})$ and $\mathrm{PGL}_{n}(\mathbb{F})$, for an arbitrary field $\mathbb{F}$, acting on the set of $1$-dimensional subspaces of $\mathbb{F}^n$. We also bound the relational complexity of all groups lying between $\mathrm{PSL}_{n}(q)$ and $\mathrm{P}Γ\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$. |
| title | The relational complexity of linear groups acting on subspaces |
| topic | Group Theory 20B15, 20G40, 03C13 |
| url | https://arxiv.org/abs/2309.16111 |