Representation dimension of some finite groups

TL;DR

This paper computes the representation dimension for specific finite groups, including p-groups and groups with particular character properties, providing explicit values and computational tools.

Contribution

It introduces methods to determine the representation dimension for certain classes of finite groups and offers GAP code for these calculations.

Findings

Computed representation dimensions for some p-groups and their products.

Identified conditions on nonlinear irreducible characters affecting representation dimension.

Provided GAP scripts for practical computation of these invariants.

Abstract

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct products, and groups with certain conditions on nonlinear irreducible characters. We also make similar computations for the smallest integer realizable as the degree of an irreducible complex faithful representation of $G$, if one exists. In the appendix, we present GAP codes to compute these numbers.