2025/10/03 by Robert Boltje, Gabriel Navarro, Boltje, Robert +1
Mathematics · #Finite Group Theory Research #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2510.03179
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.