Feit's conjecture, the canonical Brauer induction formula, and Adams operations
Robert Boltje, Gabriel Navarro
TL;DR
The paper investigates Feit’s conjecture through a conductor-anchored lens, introducing the invariant $S(G,\chi,n)$ derived from the canonical Brauer induction formula and Adams operations. It proves that $S(G,\chi,n)\ge 0$ and that $S(G,\chi,c(\chi))>0$ exactly when a representation affording $\chi$ has an eigenvalue of order $c(\chi)$, linking this to a criterion for Feit’s conjecture in the irreducible case. A core contribution is expressing $S(G,\chi,n)$ via Adams operations: $S(G,\chi,n)=\sum_{\rho\subseteq\pi(n)} (-1)^{|\rho|}(\Psi^{n(\rho)}(\chi),1_G)$, and showing this sum is nonnegative with positivity equivalent to the existence of a subgroup and a linear constituent meeting a divisibility condition on their orders. The work thus provides a computable, representation-theoretic indicator for the strong Feit condition, connects number-theoretic Lemmas to eigenvalue-order criteria, and outlines both general theory and special-case behavior, including implications for the irreducible case and practical checks via character tables and power maps.
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 $S(G,χ, c(χ))>0$.