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.
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. The Magma meataxe is currently restricted to finite fields though it is expected that this restriction will be removed in the near future.
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)
A module for constructing both characteristic zero and modular invariants of finite groups has been developed by Gregor Kemper and Allan Steel [KemSte98]. This includes a new algorithm for computing primary invariants that guarantees that the degrees of the invariants constructed are optimal (with respect to their product and their sum). Magma allows computation in invariant rings over ground fields of arbitrary characteristic. Of particular interest is the modular case, i.e., the case where the characteristic of the ground field divides the order of the group.
Permutation and matrix group actions on polynomials
Independent homogeneous invariants of a specific degree
Molien series
Primary invariants having optimal degrees (with respect to their product and then sum)
Secondary invariants of optimal degrees (using a new algorithm for the modular case)
Efficient construction of fundamental invariants
For the 4-dimensional representation of A5 over F2, optimal primary invariants (of degrees 3, 5, 8 and 12) are found in 1.5 seconds. For the cyclic matrix group of order 8 generated by the 5-dimensional Jordan form over F2, optimal primary invariants (of degrees 1, 2, 2, 4 and 8) are found in 0.6 seconds and secondary invariants with respect to these (of degrees 0, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10 and 11) are found 14.4 seconds. This last computation took many hours with algorithms prior to those implemented in Magma V2.2.
Invariant ring as a graded module over the algebra generated by the primary invariants and explicit construction of the isomorphism
Invariant ring as a polynomial algebra
Determination of algebraic relations between secondary invariants
Module syzygies between secondary invariants
Algebraic relations between invariants