Let A be an algebra over a field K and let M be a vector space over K. We say that M is a (right) A-module if for each a ∈A and m ∈M, a product ma ∈M is defined such that (m + n)a = ma + na, m(a + b) = ma + mb, m(ab) = (ma)b, m1 = m, m(ka) = (ma)k = (mk)a, for all a, b ∈A, m, n ∈M, k ∈K.
Recall that a representation of an algebra A over a field K is an algebra homomorphism of A into HomK(M, M), for some K-module M. Taking M to be an A-module, and defining a mapping φ: M -> M by φ(m) := ma (a ∈A, m ∈M) then it is an easy exercise to show that φ is a representation of A. A matrix representation of degree n of the algebra A is an algebra homomorphism of A into Mn(K), the complete matrix algebra of degree n over K. Suppose M has finite K-dimension n and choose a basis for M. If, for each a ∈A, we associate the matrix corresponding to the action of a on the basis elements, we obtain a matrix representation of A. Thus, each A-module of finite K-dimension affords a matrix representation of the algebra A. An important special case occurs when A=K[G], the group algebra of a group G. In this case the theory of A-modules coincides with the theory of group representations.
Note that in all of the above cases A has finite K-dimension.
Throughout this chapter we shall use the term A-module when referring to modules as defined above. For Magma, the A-modules M currently supported are those for which A is a matrix algebra defined over a finite field, the rational field, an algebraic number field or an Euclidean domain. However, as currently the action of A may be used only in the construction of submodules, discussion will be mainly limited to the case in which the coefficient ring of M is a field.
The situation with K[G]-modules is more complicated. The group G must be a finite permutation or matrix group or a (soluble) group defined by a polycyclic presentation. At present K must be a field. However, most algorithms depend upon having an effective version of the Meataxe for each type of field. At present these only exist for finite fields, the rational field and number fields. The functionality differs for each of these fields and so we give a brief summary of the features supported for each of the three field types.
K is a finite field: Because of the richness of the theory of modular representations a large number of intrinsics are provided and Magma has efficient algorithms for many of the calculations. The main features include:
- Construction of a K[G]-module from a variety of group actions
- Induction, restriction, inflation, etc for K[G]-modules
- Tensor products of various kinds, direct sums of K[G]-modules
- Highly optimised Meataxe for splitting modules over Fq
- Condensation used when splitting tensor products and induced modules etc
- Constituents, composition series, Jacobson radical, socle, socle series
- Submodues and quotient modules, enumerate all submodules
- Indecomposable test, direct sum decomposition, complements
- Hom spaces of various kinds, endomorphism ring, isomorphism test
- Construction of all Fq-irreducible modules for a group
- Construct the projective indecomposable modules for a group
- Cohomology and extensions of a K[G]-module
- Vertices and sources
- Bimodules
K is Q or Q(α): This is actually two cases as there is a version of the Meataxe for modules over Q and a separate version for modules over extensions of Q. For the rational field case irreducibles can be constructed from their characters provided that the degree is moderate and the group is not huge. In the number field case the degree of the field needs to be very small and the degree of modules arising in computations is only practical up to about a thousand as the splitting of such modules rapidly becomes very expensive with increasing degree. The available operations for K[G]-modules in characteristic zero are sketched below.
- Construction of a K[G]-module from a variety of group actions
- Induction, restriction, inflation, etc for K[G]-modules
- Tensor products, direct sums of K[G]-modules
- K[G]-module Longleftrightarrow representation; image and kernel
- Meataxes for splitting modules over Q and Q(α)
- Condensation used when splitting tensor products and induced modules etc
- Constituents, composition series, Jacobson radical, socle, socle series
- Indecomposable test, direct sum decomposition, complement
- Hom(M, N), endomorphism ring, isomorphism test
- Construction of an irreducible module from its rational (complex) character
- Construction of irreducible modules (for moderate sized groups)
K is a field not discussed above: As no Meataxe is available in Magma for such fields only a small number of intrinsics apply.
- Construction of a K[G]-module from a variety of group actions
- Induction, restriction, inflation etc for K[G]-modules
- Tensor products, direct sums of K[G]-modules
- K[G]-module Longleftrightarrow representation; image and kernel
Note that all of the A-module intrinsics apply to K[G]-modules.