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

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 $c(\chi)$ of an irreducible character $\chi$ 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 $\chi$ of $G$, we introduce an integer-valued invariant $S(G,\chi,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $\chi$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $\chi$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,\chi,n)$ is always non-negative, and it is positive if and only if a representation affording $\chi$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $\chi$ if and only if $S(G,\chi, c(\chi))>0$.