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

TL;DR

This paper investigates a conjecture about sharp characters in finite groups, providing counterexamples and proving the conjecture under specific conditions involving normalized pairs and irrational values.

Contribution

It offers counterexamples to a 1988 conjecture and proves the conjecture for normalized pairs with at least one irrational value in the set L.

Findings

Counterexamples to Cameron and Kiyota's conjecture.

Proof of the conjecture for normalized pairs with irrational values.

Conditions under which the inner product is uniquely determined.

Abstract

Let $ \chi $ be a virtual (generalized) character of a finite group $ G $ and $ L=L(\chi)$ be the image of $ \chi $ on $ G-\lbrace 1 \rbrace $. 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 $|L|\geq 2$, 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.