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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. Also the GAP 3 share package AREP offers possibilities to calculate with representations (which are of interest e.g. for applications to signal processing).

Efficient methods are available for special classes of groups, e. g. in the GAP 3 share package CHEVIE which allows one to work with generic characters of groups of Lie type and Hecke algebras. See also the related GAP 3 share package Specht.

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, vector enumeration (in GAP 3), 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.
  • The GAP Library of Tables of Marks provides these for many nearly simple and related groups.