GAP (repsn) - Chapter 1: Introduction

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1 Introduction

### 1 Introduction

This manual describes the Repsn package for computing matrix representations in characteristic zero of finite groups. Most of the functions in Repsn have been written according to the algorithm described in the author's Ph.D thesis [DA03] and [DD10] (see [DA05]).

For constructing representations of simple groups and their covers we use the algorithm described in [Dix93]. To use this algorithm for constructing a representation of a group \(G\) affording an irreducible character \(\chi\) of \(G\), we need to have a subgroup \(H\) of \(G\) such that the restriction of \(\chi\) to \(H\) has a linear constituent with multiplicity one. In this case we say \(H\) is a character subgroup relative to \(\chi\) (or a \(\chi\)-subgroup). A \(\chi\)-subgroup for each irreducible character \(\chi\) of degree less than 100 of simple groups and their covers are listed in [DA06] and [DA07].

All Repsn functions are written entirely in the GAP language. It is proved in [DA05] and [DD10] that the algorithm is correct for any group with a character of degree less than 100. Indeed, if the group is solvable, there is no restriction on the character degree. In practice the program is quite fast when the degree is small, but can be very slow when it is necessary to call one of the subprograms which extend irreducible representations. In the latter case the number of element wise operations required to extend a representation of degree \(d\) is proportional to \(d^6\).

Repsn is implemented in the GAP language, and runs on any system supporting GAP4. The Repsn package is loaded into the current GAP session with the command

     gap> LoadPackage( "repsn" );

(see section Loading a GAP Package in the GAP Reference Manual).

Please report any bugs or other issues you might encounter via the Repsn issue tracker at https://github.com/gap-packages/repsn/issues.

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