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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.03179 |
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| _version_ | 1866912626533990400 |
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| author | Boltje, Robert Navarro, Gabriel |
| author_facet | Boltje, Robert Navarro, Gabriel |
| contents | This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(χ)$ of an irreducible character $χ$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $χ$ of $G$, we introduce an integer-valued invariant $S(G,χ,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $χ$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $χ$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,χ,n)$ is always non-negative, and it is positive if and only if a representation affording $χ$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $χ$ if and only if $S(G,χ, c(χ))>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03179 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Feit's conjecture, the canonical Brauer induction formula, and Adams operations Boltje, Robert Navarro, Gabriel Representation Theory Group Theory 20C15 This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(χ)$ of an irreducible character $χ$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $χ$ of $G$, we introduce an integer-valued invariant $S(G,χ,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $χ$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $χ$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,χ,n)$ is always non-negative, and it is positive if and only if a representation affording $χ$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $χ$ if and only if $S(G,χ, c(χ))>0$. |
| title | Feit's conjecture, the canonical Brauer induction formula, and Adams operations |
| topic | Representation Theory Group Theory 20C15 |
| url | https://arxiv.org/abs/2510.03179 |