On common zeros of characters of finite groups

2024/09/23 by Mark L. Lewis, Lucia Morotti, Lewis, Mark L. +7
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #Representation Theory (math.RT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2409.14811

openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let G be a finite group, and let Irr(G) denote the set of the irreducible complex characters of G. An element g∈ G is called a vanishing element of G if there exists χ\inIrr(G) such that χ(g)=0 (i.e., g is a zero of χ) and, in this case, the conjugacy class g^G of g in G is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group G such that every non-linear χ\inIrr(G) vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.

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