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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.09277 |
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| _version_ | 1866916029612949504 |
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| author | Lawrence, Thomas Semeraro, Jason |
| author_facet | Lawrence, Thomas Semeraro, Jason |
| contents | A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the determinant of $X \overline{X}$ (the square of the volume of this lattice) with the product of the orders of $S$-centralisers of fully $\mathcal{F}$-centralised $\mathcal{F}$-class representatives. This statement is exactly column orthogonality for the character table of $S$ when $\mathcal{F}=\mathcal{F}_S(S)$. We prove the conjecture when $\mathcal{F}=\mathcal{F}_S(G)$ is realised by some finite group $G$ with Sylow $p$-subgroup $S$, and for all simple fusion systems when $|S| \le p^4$. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of $\mathcal{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On character tables for fusion systems Lawrence, Thomas Semeraro, Jason Representation Theory 20C15, 19A22 A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the determinant of $X \overline{X}$ (the square of the volume of this lattice) with the product of the orders of $S$-centralisers of fully $\mathcal{F}$-centralised $\mathcal{F}$-class representatives. This statement is exactly column orthogonality for the character table of $S$ when $\mathcal{F}=\mathcal{F}_S(S)$. We prove the conjecture when $\mathcal{F}=\mathcal{F}_S(G)$ is realised by some finite group $G$ with Sylow $p$-subgroup $S$, and for all simple fusion systems when $|S| \le p^4$. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of $\mathcal{F}$. |
| title | On character tables for fusion systems |
| topic | Representation Theory 20C15, 19A22 |
| url | https://arxiv.org/abs/2510.09277 |