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arXiv · 2510.03179

Feit's conjecture, the canonical Brauer induction formula, and Adams operations

Robert Boltje·Gabriel Navarro
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Abstract

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 $S(G,χ, c(χ))>0$.

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BibTeXRIS

Robert Boltje, Gabriel Navarro. 2025-10-03. Feit's conjecture, the canonical Brauer induction formula, and Adams operations. https://arxiv.org/abs/2510.03179

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