Partial character tables for $\mathbb{Z}_\ell$-spetses
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| Format: | Preprint |
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2025
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| _version_ | 1866911050867146752 |
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| author | Kessar, Radha Malle, Gunter Semeraro, Jason |
| author_facet | Kessar, Radha Malle, Gunter Semeraro, Jason |
| contents | Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$.
Secondly, when $\ell > 2$ is a very good prime for ${\mathbb{G}}$, the Weyl group $W$ of ${\mathbb{G}}$ has order coprime with $\ell$, and $q\equiv1\pmod\ell$ we introduce a formula for the values of characters in the principal block of ${\mathbb{G}}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality.
In both cases we formulate and provide evidence for several conjectures concerning the proposed values. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partial character tables for $\mathbb{Z}_\ell$-spetses Kessar, Radha Malle, Gunter Semeraro, Jason Representation Theory Group Theory 20C08, 20C20, 20F55, 16G30, 20D20, 55R35 Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$. Secondly, when $\ell > 2$ is a very good prime for ${\mathbb{G}}$, the Weyl group $W$ of ${\mathbb{G}}$ has order coprime with $\ell$, and $q\equiv1\pmod\ell$ we introduce a formula for the values of characters in the principal block of ${\mathbb{G}}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality. In both cases we formulate and provide evidence for several conjectures concerning the proposed values. |
| title | Partial character tables for $\mathbb{Z}_\ell$-spetses |
| topic | Representation Theory Group Theory 20C08, 20C20, 20F55, 16G30, 20D20, 55R35 |
| url | https://arxiv.org/abs/2507.08502 |