This section describes facilities in Magma that relate to the representation theory of groups and associative algebras. The main topics considered include:
Modules over an algebra
K[G]-modules
Representations of groups
Character theory
Invariant theory
We consider a module whose elements are n-tuples over a field K with an action given by a matrix representation of an associative algebra A. We will refer to these modules as A-modules. These include K[G]-modules.
The four fundamental algorithms for computational module theory are echelonization, the spinning algorithm, the meataxe algorithm and an algorithm for Hom(U,V). For the important case of modules over finite fields, different representations of vector arithmetic, depending upon the field, have been implemented.
Creation from the matrix representation of an associative algebra.
Creation from group actions of different kinds
Permutation module of a group corresponding to its action on the cosets of a subgroup
K[G]-modules corresponding to actions of a permutation or matrix group on a polynomial ring.
Extension and restriction of the field of scalars
Direct sum
Tensor product, symmetric square, exterior square (K[G]-modules only)
Dual (K[G]-modules only)
Induction and restriction (K[G]-modules only)
All irreducible K[G]-modules of a finite soluble group where K is a finite field or field of characteristic zero
All irreducible K[G]-modules of a finite group where K is restricted to be a finite field or the rational field.
Submodules via the spinning algorithm
Membership of a submodule
Basis operations
Sum and intersection of submodules
Quotient modules
Splitting a reducible module (Holt-Rees Meataxe)
Testing a module for irreducibility, absolute irreducibility
Centralizing algebra of an irreducible module
Composition series, composition factors, constituents
Maximal and minimal submodules
Jacobson radical, socle
Socle series
Existence of a complement of a submodule
One complement, all complements of a direct summand
Testing modules for indecomposability; indecomposable components
Submodule lattice for modules over a finite field
The Magma algorithm for splitting modules (the Meataxe algorithm) is a deterministic version of the Holt-Rees algorithm and is capable of splitting modules over GF(2) having dimension up to at least 20 000.
Since V2.16, a new Meataxe algorithm is used for splitting general A-modules, where A is a finite dimensional matrix algebra defined over the rational field. This yields an effective algorithm for decomposing a module into indecomposable summands. If the module is a G-module for some group G, extensive use is also made of character theory. Representations associated with characters having non-trivial Schur indices are properly handled. The difficult problem of splitting homogeneous modules (direct sums of the same indecomposable) is handled by decomposing the endomorphism ring of the module via a maximal order. Modules having dimensions in the several hundreds are routinely split into indecomposable modules.
Dixon's method to compute the representation from a character
For soluble groups: Brückner's method to compute all absolutely irreducible representations
For general finite groups: a new algorithm of Steel constructs all irreducible representations over the rational field
Given an absolutely irreducible G-module over a number field or finite field, write it over any related field possible. Find a smallest field of definition
Restriction of scalars: use the module structure of the coefficient ring to obtain reducible representations over any subfield
For representaions over ℚ, find isomorphic integral representations
For representations over number fields, decide if the representation can be made integral. Find all classes of integrally equivalent representations of an absolutely irreducible one.
Compute modular representations from representations over number fields at any prime ideal
Try to find "nicer" versions of a representation
Special functionality for representations of a symmetric group concentrates on characters as indexed by partitions of weight the degree of the group.
Integral, seminormal and orthogonal representations of a permutation.
Values of a character of a symmetric group indexed by a partition on a permutation.
Characters of symmetric groups corresponding to partitions.
Values of a character of an alternating group indexed by a partition on a permutation.
Characters of symmetric groups corresponding to partitions.
The character theory machinery is currently restricted to characters defined over the complex field.
Definition of class functions
Construction of permutation characters
Arithmetic on class functions: sum, difference, tensor product
Frobenius-Schur indicator
Norm, order, kernel, centre of a character
Properties: generalized character, character, irreducible, faithful, linear
Induction and restriction of a character
Decomposition of a tensor power: orthogonal components, symmetric components
Action of a group on the characters of a normal subgroup
Decomposition of characters
Class matrix, structure constants for centre of group algebra
Table of ordinary irreducible characters (Dixon-Schneider algorithm, Unger's algorithm)