Representations induced from a normal subgroup of prime index

TL;DR

This paper investigates how induced representations from a normal subgroup of prime index decompose over fields that are not algebraically closed, revealing five possible decomposition types.

Contribution

It extends existing results by classifying the decomposition of induced representations over non-algebraically closed fields, identifying five distinct cases.

Findings

Induced representations are either irreducible or sum of p inequivalent irreducibles over algebraically closed fields.

Over non-algebraically closed fields, five possible decomposition types are identified.

The classification depends on the properties of the base field and the subgroup.

Abstract

Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $\eta$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of characteristic either $0$ or prime to $|G|$, then the induced representation $\eta \uparrow^{G}_{H}$ is either irreducible or a direct sum of $p$ pairwise inequivalent irreducible representations. In this paper, we show that if $F$ is not assumed algebraically closed field, then there are five possibilities in the decomposition of induced representation into irreducible representations.