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

Abstract

Let $\chi$ be a virtual (generalized) character of a finite group $G$ and $L=L(\chi )$ be the image of $\chi$ on $G-\{1\}$. The pair $(G,\chi )$ is said to be sharp of type $L$ if $|G|={\prod }_{l\in L}(\chi (1)-l)$. If the principal character of $G$ is not an irreducible constituent of $\chi$, the pair $(G,\chi )$ 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 $(G,\chi )$ is sharp and $|L|\ge 2$, then the inner product $(\chi ,\chi {)}_G$ is uniquely determined by $L$. We then prove that this conjecture is true in the case that $(G,\chi )$ is normalized, $\chi$ is a character of $G$, and $L$ contains at least an irrational value.