A conjecture of Cameron and Kiyota on sharp characters with prescribed values
Aliréza Abdollahi, Javad Bagherian, Maryam Khatami, Z. Shahbazi, R. Sobhani · Communications in Algebra · 2022
Let χ be a virtual (generalized) character of a finite group G and L=L(χ) be the image of χ on G−{1}. The pair (G,χ) is said to be sharp of type L or L-sharp if |G|=∏l∈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 L-sharp and |L|≥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.