Characters of prime degree

TL;DR

This paper investigates the structure of products of irreducible characters of prime degree in finite nilpotent groups, revealing conditions under which their products decompose into irreducibles or linear combinations.

Contribution

It establishes new criteria for the decomposition of products of prime degree irreducible characters in finite nilpotent groups.

Findings

If the product is a multiple of an irreducible, then specific structural conditions hold.

Otherwise, the product decomposes into at least (p+1)/2 distinct irreducible characters.

Results deepen understanding of character theory in finite nilpotent groups.

Abstract

Let $G$ be a finite nilpotent group, $\chi$ and $\psi$ be irreducible complex characters of $G$ of prime degree. Assume that $\chi(1)=p$. Then either the product $\chi\psi$ is a multiple of an irreducible character or $\chi\psi$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.