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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.09375 |
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| _version_ | 1866914693307695104 |
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| author | Gómez-Gonzáles, Claudio Sutherland, Alexander J. Wolfson, Jesse |
| author_facet | Gómez-Gonzáles, Claudio Sutherland, Alexander J. Wolfson, Jesse |
| contents | Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a cylic group; an alternating group; a simple factor of a Weyl group of type $E_6$, $E_7$, or $E_8$; or $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$. In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) $\operatorname{RD}_k^{\leq d}$-versality, which we connect to the existence of "special points" on varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09375 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups Gómez-Gonzáles, Claudio Sutherland, Alexander J. Wolfson, Jesse Algebraic Geometry Group Theory Number Theory 14L30 (Primary), 13A50, 20C25, 20C34 (Secondary) Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a cylic group; an alternating group; a simple factor of a Weyl group of type $E_6$, $E_7$, or $E_8$; or $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$. In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) $\operatorname{RD}_k^{\leq d}$-versality, which we connect to the existence of "special points" on varieties. |
| title | Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups |
| topic | Algebraic Geometry Group Theory Number Theory 14L30 (Primary), 13A50, 20C25, 20C34 (Secondary) |
| url | https://arxiv.org/abs/2310.09375 |