We summarise the contents of the package and provide references for the relevant algorithms.
(a) Irreducibility and absolutely irreducibility for G-modules; isomorphism testing for irreducible G-modules; see Holt and Rees IsIrreducible for GModules, IsAbsolutelyIrreducible, HomGModule, IsomorphismGModule, CompositionFactors, FieldGenCentMat, MinimalSubGModules.
(b) Decide whether a matrix group has certain decompositions with respect to a normal subgroup; see Holt, Leedham-Green, O'Brien and Rees [6]. The corresponding functions are described in IsSemiLinear, SmashGModule, SemiLinearDecomposition, TensorProductDecomposition, SymTensorProductDecomposition, and ExtraSpecialDecomposition.
(c) Decide whether a matrix group is primitive; see Holt, Leedham-Green, O'Brien and Rees [7]. The corresponding functions are described in IsPrimitive for GModules, MinBlocks.
(d) Decide whether a given group contains a classical group in its natural representation. Here we provide access to the algorithms of Celler and Leedham-Green [3] and those of Niemeyer and Praeger [11, 12]. The corresponding function is described in RecogniseClassical, the associated lower-level functions in RecogniseClassicalCLG and RecogniseClassicalNP.
(e) A constructive recognition process for the special linear group developed by Celler and Leedham-Green [4] and described in ConstructivelyRecogniseClassical.
(e) Random element selection; see Celler, Leedham-Green, Murray, Niemeyer and O'Brien [1]. The corresponding functions are described in PseudoRandom, InitPseudoRandom.
(f) Matrix order calculation; see Celler and Leedham-Green [2]. The corresponding functions are described in OrderMat -- enhanced.
(g) Base point selection for the Random Schreier-Sims algorithm for matrix groups; see Murray and O'Brien [10]. The corresponding function is described in PermGroupRepresentation.
(h) Decide whether a matrix group preserves a tensor decomposition; see Leedham-Green and O'Brien [8, 9]. The corresponding function is described in IsTensor.
(i) Recursive exploration of reducible groups; see Pye [13]. The corresponding function is described in RecogniseMatrixGroup.
The algorithms make extensive use of Aschbacher's classification of the maximal subgroups of the general linear group. Possible classes of subgroups mentioned below refer to this classification; see [14, 15] for further details.
In order to access the functions, you must use the command
RequirePackage to load them.
gap> RequirePackage("matrix");
GAP 3.4.4