GAP (SL2Reps) - Chapter 1: Introduction

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1 Introduction
1.1 Installation
1.2 Usage

### 1 Introduction

This package, `SL2Reps`, provides methods for constructing and testing matrix presentations of the representations of \(\mathrm{SL}_2(\mathbb{Z})\) whose kernels are congruence subgroups of \(\mathrm{SL}_2(\mathbb{Z})\).

Irreducible representations of prime-power level are constructed individually by using the Weil representations of quadratic modules, and from these a list of all representations of a given degree or level can be produced. Each representation is represented by a pair \((S,T)\), where \(S\) is a symmetric, unitary matrix and \(T\) is a diagonal matrix of finite order; this format is designed for the study of modular tensor categories in particular.

#### 1.1 Installation

To install `SL2Reps`, first download it from https://snw-0.github.io/sl2-reps/, then place it in the `pkg` subdirectory of your GAP installation (or in the `pkg` subdirectory of any other GAP root directory, for example one added with the `-l` argument).

`SL2Reps` is then loaded with the GAP command

`gap> LoadPackage( "SL2Reps" );`

#### 1.2 Usage

Specific irreducible representations may be constructed via the methods in Chapter 3, while lists of irreducible representations with a given degree or level may be constructed with those in Chapter 4.

This package uses an `InfoClass`, `InfoSL2Reps`. It may be set to `0` (silent), `1` (info), or `2` (verbose). To change it, use

`gap> SetInfoLevel( InfoSL2Reps, k );`

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