Homogeneous products of characters

TL;DR

This paper investigates conditions under which the product of faithful irreducible characters in solvable groups is irreducible, proving a special case of a conjecture and establishing results for p-groups.

Contribution

It proves a special case of a conjecture relating to the irreducibility of character products in solvable groups, including p-groups.

Findings

If the product of two faithful irreducible characters is a scalar multiple of an irreducible character, then both characters vanish outside the center.

The conjecture holds true for p-groups.

The paper extends Isaacs' conjecture to a broader class of character products.

Abstract

I. M. Isaacs has conjectured (see \cite{isa00}) that if the product of two faithful irreducible characters of a solvable group is irreducible, then the group is cyclic. In this paper we prove a special case of the following conjecture, which generalizes Isaacs conjecture. Suppose that $G$ is solvable and that $\psi,\phi\in\Irr(G)$ are faithful. If $\psi \phi=m\chi$ where $m$ is a positive integer and $\chi \in \Irr(G)$ then $\psi$ and $\phi$ vanish on $G- Z(G)$. In particular we prove that the above conjecture holds for $p$-groups.