Character values and conductors of low-rank groups of Lie type
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918449413881856 |
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| author | Herbig, Christopher Hung, Nguyen N. |
| author_facet | Herbig, Christopher Hung, Nguyen N. |
| contents | Let $χ$ be a complex irreducible character of a finite group $G$. The conductor of $χ$, denoted $c(χ)$, is the smallest positive integer $n$ such that $χ(x)\in \mathbb{Q}(\exp({2πi/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(χ)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(χ)=c(χ(g))$. In some quasisimple cases, we further prove that the field of values \(\mathbb{Q}(χ)\) is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10888 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Character values and conductors of low-rank groups of Lie type Herbig, Christopher Hung, Nguyen N. Representation Theory Group Theory Number Theory 20C15, 11R18, 20C33, 11R20 Let $χ$ be a complex irreducible character of a finite group $G$. The conductor of $χ$, denoted $c(χ)$, is the smallest positive integer $n$ such that $χ(x)\in \mathbb{Q}(\exp({2πi/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(χ)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(χ)=c(χ(g))$. In some quasisimple cases, we further prove that the field of values \(\mathbb{Q}(χ)\) is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups. |
| title | Character values and conductors of low-rank groups of Lie type |
| topic | Representation Theory Group Theory Number Theory 20C15, 11R18, 20C33, 11R20 |
| url | https://arxiv.org/abs/2604.10888 |