Restriction of characters and products of characters

TL;DR

This paper explores how the product of irreducible characters in finite p-groups can be expressed as sums of multiple irreducibles, extending previous results that required faithfulness of characters.

Contribution

It generalizes earlier theorems by showing that, without faithfulness, products of irreducible characters can decompose into any number of irreducibles, using character restriction decompositions.

Findings

Products of irreducible characters can be nontrivial sums of any number of irreducibles.

Previous bounds hold only for faithful characters, not in general.

Construction of examples for any number of irreducibles in the product.

Abstract

Let G be a finite p-group, for some prime p, and $\psi, \theta \in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $\psi,\theta$ are faithful characters, then the product $\psi\theta$ 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 $\Psi$ and $\Theta$ such that $\Psi\Theta$ 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.