Representations and Characters of Groups

Group representations over fields of characteristic zero are mainly investigated via their characters. GAP provides methods for computing the irreducible characters of a given finite group, either automatically or interactively by character theoretic means. It also provides many functions for deducing group theoretic properties from character tables.

The computation of the irreducible representations themselves is possible for not too large groups (see e. g. the function ‘IrreducibleRepresentations’ in the reference manual section Computing the Irreducible Characters of a Group). The package Repsn provides methods for the construction of characteristic zero representations of finite groups.

Modular representations (i. e., over fields whose characteristic divides the group order) can be studied via Brauer characters or by explicit calculations with matrices representing the generators of the group in question, using MeatAxe methods, and condensation techniques.

Several GAP data libraries are related to representations and characters.

• The GAP Character Table Library gives access to ordinary and modular character tables of many nearly simple groups and of related groups such as their maximal subgroups.
• The Atlas of Group Representations gives access to many permutation and matrix representations of many nearly simple and related groups.