Center-preserving irreducible representations of finite groups

TL;DR

This paper investigates conditions under which certain irreducible representations of finite groups preserve the center structure when restricted to subgroups, establishing a link between subgroup representations and the larger group's representations.

Contribution

It proves that if a subgroup has a faithful irreducible representation, then an induction to the larger group contains a center-preserving irreducible component, linking subgroup and group representations.

Findings

Existence of center-preserving irreducible components in induced representations

Characterization of when a subgroup has a faithful irreducible representation

Examples illustrating the sharpness of the main results

Abstract

Given finite groups $H \leq G$, a representation $\sigma$ of $G$ is called center-preserving on $H$ if the only elements of $H$ that become central under $\sigma$ are those that were already central in $G$. We prove that if $H$ has a faithful irreducible representation $\rho$, then at least one of the irreducible components of the induction $\operatorname{Ind}_H^G(\rho)$ is center-preserving on $H$. In consequence, $H$ has a faithful irreducible representation if and only if every finite group $G$ containing $H$ as a subgroup has an irreducible representation whose restriction to $H$ is faithful, and which is center-preserving on $H$. In addition, we give examples illustrating the sharpness of the statement, and discuss the connection with projective representations.