A counterexample to a conjecture of A. R. Miller

2025/10/12 by Navarro, Gabriel, Sambale, Benjamin
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2510.10734

Abstract

Let χ be an irreducible character of a finite group G. A. R. Miller conjectured that the proportion of elements g∈ G such that χ(g) is zero or a root of unity is at least 1/2. We construct a character of a perfect group of order 69120 such that this proportion is 511/1152.

Citations

• Denseness results for zeros and roots of unity in character tables
• Short character values on large conjugacy classes
• New Lower Bounds for the Schur-Siegel-Smyth Trace Problem

Related

• Zeros and roots of unity in character tables
• Denseness results for zeros and roots of unity in character tables
• Orders of elements and zeros and heights of characters in a finite group
• Finite groups with a small proportion of vanishing elements
• On the average character degree of finite groups