Representations of $GL_2(\Fq)$ and $SL_2(\Fq)$, and some remarks about $GL_n(\Fq)$

TL;DR

This paper provides a comprehensive overview of the representation theory of $GL_2(q)$ and $SL_2(q)$, including insights into the more general case of $GL_n(q)$, aimed at clarifying the structure and properties of these representations.

Contribution

It offers a self-contained exposition of the representation theory for $GL_2$ and $SL_2$ over finite fields, with preliminary remarks on the broader $GL_n$ case.

Findings

Detailed classification of representations of $GL_2(q)$ and $SL_2(q)$

Insights into the structure of $GL_n(q)$ representations

Clarification of the theory's foundational aspects

Abstract

The goal of these notes is to give a self-contained account of the representation theory of $GL_2$ and $SL_2$ over a finite field, and to give some indication of how the theory works for $GL_n$ over a finite field.