Representation Rings of Fusion Systems and Brauer Characters
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912023933091840 |
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| author | Lawrence, Thomas |
| author_facet | Lawrence, Thomas |
| contents | Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$. We study the ring $R(\mathcal{F})$ of $\mathcal{F}$-stable characters by exploiting a new connection to the modular characters of a finite group $G$ with $\mathcal{F} = \mathcal{F}_S(G)$. We utilise this connection to find the rank of the $\mathcal{F}$-stable character ring over fields with positive characteristic. We use this theory to derive a decomposition of the regular representation for a fixed basis $B$ of the ring of complex $\mathcal{F}$-stable characters and give a formula for the absolute value of the determinant of the $\mathcal{F}$-character table with respect to $B$ (the matrix of the values taken by elements of $B$ on each $\mathcal{F}$-conjugacy class) for a wide class of saturated fusion systems, including all non-exotic fusion systems, and prove this value squared is a power of $p$ for all saturated fusion systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation Rings of Fusion Systems and Brauer Characters Lawrence, Thomas Representation Theory Group Theory 20C99 (Primary) 20D20, 20C20 (Secondary) Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$. We study the ring $R(\mathcal{F})$ of $\mathcal{F}$-stable characters by exploiting a new connection to the modular characters of a finite group $G$ with $\mathcal{F} = \mathcal{F}_S(G)$. We utilise this connection to find the rank of the $\mathcal{F}$-stable character ring over fields with positive characteristic. We use this theory to derive a decomposition of the regular representation for a fixed basis $B$ of the ring of complex $\mathcal{F}$-stable characters and give a formula for the absolute value of the determinant of the $\mathcal{F}$-character table with respect to $B$ (the matrix of the values taken by elements of $B$ on each $\mathcal{F}$-conjugacy class) for a wide class of saturated fusion systems, including all non-exotic fusion systems, and prove this value squared is a power of $p$ for all saturated fusion systems. |
| title | Representation Rings of Fusion Systems and Brauer Characters |
| topic | Representation Theory Group Theory 20C99 (Primary) 20D20, 20C20 (Secondary) |
| url | https://arxiv.org/abs/2409.03007 |