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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.03976 |
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| _version_ | 1866909859527524352 |
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| author | Robinson, Geoffrey R. |
| author_facet | Robinson, Geoffrey R. |
| contents | We consider the generalized character $Ψ_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y \in G$ is the number of $p$-elements of $C_{G}(y)$. We conjecture that this is always a character, and may be afforded by a projective $RG$-module, where $R$ is an appropriate complete discrete valuation ring whose residue field has characteristic $p$. We examine a number of case where this is the case, and consider consequences for the representation theory and character theory of $G$ when this conjecture is known to hold. In particular, we prove, among other things, that the conjecture is valid for all primes $p$ in the case that $G \cong {\rm PSL}(2,q)$ or ${\rm SL}(2,q)$ for every prime power $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalized character related to the local structure and representation theory of a finite group Robinson, Geoffrey R. Representation Theory Group Theory Primary 20C20 We consider the generalized character $Ψ_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y \in G$ is the number of $p$-elements of $C_{G}(y)$. We conjecture that this is always a character, and may be afforded by a projective $RG$-module, where $R$ is an appropriate complete discrete valuation ring whose residue field has characteristic $p$. We examine a number of case where this is the case, and consider consequences for the representation theory and character theory of $G$ when this conjecture is known to hold. In particular, we prove, among other things, that the conjecture is valid for all primes $p$ in the case that $G \cong {\rm PSL}(2,q)$ or ${\rm SL}(2,q)$ for every prime power $q$. |
| title | A generalized character related to the local structure and representation theory of a finite group |
| topic | Representation Theory Group Theory Primary 20C20 |
| url | https://arxiv.org/abs/2505.03976 |