Various Representation Dimensions associated with a Finite Group

TL;DR

This survey reviews various notions of dimensions related to finite groups, including embedding degree, character degree, permutation representation degree, and essential dimension, highlighting recent progress and open problems.

Contribution

It consolidates different dimension concepts associated with finite groups and discusses recent advances and unresolved issues in understanding these notions.

Findings

Summarizes progress in understanding various group dimensions.

Identifies key open problems in the area.

Connects different notions of dimensions for finite groups.

Abstract

To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear Groups and permutation groups. These are embedding degree, minimal faithful irreducible character degree, minimal faithful permutation representation degree, minimal faithful quasi-permutation representation degree and essential dimension. We briefly present the progress in understanding these notions and the related problems.