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
Basic Algebras
Representations of groups
Character theory
Cohomology and group extensions
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.
Creation of bimodules from group actions and as permutation modules.
Extraction of either one-sided module from a bimodule.
Extension and restriction of the field of scalars
Direct sum
Tensor product, symmetric square, exterior square (K[G]-modules only)
Tensor products of bimodules.
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.
Construction of projective indecomposable K[G]-modules where K is a finite field.
Construction of a vertex and source for an indecomposable K[G]-modules where K is a finite field.
Irreducible modules over finite fields may be constructed for fairly large groups. For the Rudvalis simple group (order 145926144000), representations over GF(2) of dimensions 28, 376, 1246, and 7280 are found in 22 seconds.
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: an 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
A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one. In the literature such an algebra is known as a "split" basic algebra. Every finite dimensional algebra is Morita equivalent to a basic algebra, meaing that the algebra and its basic algebra have equivalent module categories and hence the same representation theory. So, for example, a cohomology calculation involving modules over an algebra is often most easily done by condensing to the basic algebra. Magma has the capabilility of constructing the basic algebra of a matrix algebra defined over a finite field. In some cases it is necessary to extend the field in order to split the irreducible modules over the matrix algebra. The basic algebra type in Magma is optimized for the purposes of doing homological calculations.
Creation from a sequence of projective modules and a path tree for each module
Creation of the split basic algebra over a finite field corresponding to a matrix algebra or endomorphism algebra.
Creation of the basic algebra corresponding to a Schur algebra or Hecke algebra over a finite field.
Creation of the basic algebra corresponding to the group algebra of a p-group over GF(p).
Arithmetic
Extension and restriction of the coefficient ring
Tensor product
Opposite algebra
Construction of modules over basic algebras
Submodules, quotient modules, radicals and socles
Algebra considered as a right regular module over itself
The space Hom(M,N) of all homomorphisms (all projective homomorphisms) from module M to module N
Pushouts and pullbacks with respect to module homomorphisms
Projective resolution as a complex of modules; projective covers
Injective resolution as a complex of modules; injective hulls
Calculation of the Ext algebra of a basic algebra
Restriction and inflation for basic algebras of p-groups
Cohomology ring of the unique simple module k for the basic algebra of a p-group
Calculation of A∞ algebras structures on cohomology rings
The general character theory machinery is currently restricted to characters defined over the complex field. The algorithms for computing a group character table can compute tables for groups with order up to 1020 and 8000 characters. For instance GU(3,19) with order 338,779,728,000 and 7640 characters has its character table computed in 133000 seconds.
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)
Partition of character table into blocks modulo p.
Schur indices of characters
Representation corresponding to an irreducible character
For the Coxeter group H4, the degree 48 irreducible character is faithful and rational, with indicator 1 and Schur index 2. A representation affording this character is found in 2 seconds. The representation is over a real number field of degree 2 over the rationals. Constructing representations corresponding to all 34 irreducible characters takes 12 seconds.
Special functionality for representations of a symmetric group concentrates on characters as indexed by partitions of weight the degree of the group.
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.
Integral, seminormal and orthogonal representations of a permutation.
The cohomology functions are designed to provide a flexible set of tools for computing with first and second cohomology groups of any type of finite group acting on any reasonable module, including a module defined by an action on an arbitrary finitely generated abelian group. First (but not second) cohomology groups can also be calculated for infinite groups defined by a finite presentation.
Find the dimensions of the first and second cohomology groups.
Construct the first and second cohomology groups.
Corestrictions and lifting of cocycles.
Given an element of second cohomology group H2(G,M), construct, as a finitely presented group, the corresponding extension of G by M.
Construction of all distinct extensions of G by M.
Construct, as finitely presented groups, p-multiplicator and p-cover of a permutation group.
Functions are provided for computing the first cohomology group of a finite group with coefficients in a finite (not necessarily abelian) group.
Construction of Γ-groups, and quotients by Γ-invariant normal subgroups.
Construction of 1-cocycles and testing for equivalence.
Construction of the first cohomology H1(Γ,A) corresponding to a Γ-group A.