Restriction of characters and products of characters

Abstract

Let G be a finite p-group, for some prime p, and $ψ, θ\in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $ψ,θ$ are faithful characters, then the product $ψθ$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $Ψ$ and $Θ$ such that $ΨΘ$ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.