Rational conjugacy classes and rational characters for some finite simple groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908285048717312 |
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| author | Kaur, Dilpreet Panja, Saikat |
| author_facet | Kaur, Dilpreet Panja, Saikat |
| contents | If $G$ is a finite group, an irreducible complex-valued character $χ$ is called rational if $χ(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $χ$, the value $χ(x^G)$ is rational. We prove that for $q$, a power of prime, the group $\mathrm{PSL}_2(q)$ has same number of rational characters and rational conjugacy classes. Furthermore, we verify that this equality holds for all finite simple groups whose character tables appear in the $\textit{ATLAS of Finite Groups}$, except for the Tits group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational conjugacy classes and rational characters for some finite simple groups Kaur, Dilpreet Panja, Saikat Group Theory Representation Theory 20C15, 20D06, 20E45 If $G$ is a finite group, an irreducible complex-valued character $χ$ is called rational if $χ(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $χ$, the value $χ(x^G)$ is rational. We prove that for $q$, a power of prime, the group $\mathrm{PSL}_2(q)$ has same number of rational characters and rational conjugacy classes. Furthermore, we verify that this equality holds for all finite simple groups whose character tables appear in the $\textit{ATLAS of Finite Groups}$, except for the Tits group. |
| title | Rational conjugacy classes and rational characters for some finite simple groups |
| topic | Group Theory Representation Theory 20C15, 20D06, 20E45 |
| url | https://arxiv.org/abs/2503.20452 |