A conjecture of Cameron and Kiyota on sharp characters with prescribed values

Abstract

Let $ χ$ be a virtual (generalized) character of a finite group $ G $ and $ L=L(χ)$ be the image of $ χ$ on $ G-\lbrace 1 \rbrace $. The pair $ (G, χ) $ is said to be sharp of type $ L $ if $|G|=\prod _{ l \in L} (χ(1) - l) $. If the principal character of $G$ is not an irreducible constituent of $χ$, the pair $(G,χ)$ is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in $1988$. This conjecture states that if $(G,χ)$ is sharp and $|L|\geq 2$, then the inner product $(χ,χ)_G$ is uniquely determined by $ L $. We then prove that this conjecture is true in the case that $(G,χ) $ is normalized, $χ$ is a character of $ G $, and $ L $ contains at least an irrational value.