Category of representations
In representation theory, the category of representations of some algebraic structure has the representations of as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects.
The Tannakian formalism gives conditions under which a group G may be recovered from the category of representations of it together with the forgetful functor to the category of [vector spaces].
The Grothendieck ring of the category of finite-dimensional representations of a group G is called the representation ring of G.
Definitions
Depending on the types of representations one wants to consider, it is typical to use slightly different definitions.For a finite group and a field, the category of representations of over has Objects: Pairs of vector spaces over and representations of on that vector spaceMorphisms: Equivariant mapsComposition: The composition of equivariant mapsIdentities: The identity function.
The category is denoted by or.
For a Lie group, one typically requires the representations to be smooth or admissible. For the case of a Lie algebra, see Lie algebra representation. See also: category O.
The category of modules over the group ring
There is an isomorphism of categories between the category of representations of a group over a field and the category of modules over the group ring, denoted -Mod.Category-theoretic definition
Every group can be viewed as a category with a single object, where morphisms in this category are the elements of and composition is given by the group operation; so is the automorphism group of the unique object. Given an arbitrary category ', a representation of in ' is a functor from to '. Such a functor sends the unique object to an object say ' in ' and induces a group homomorphism ; see Automorphism group#In category theory for more. For example, a -set is equivalent to a functor from to Set, the category of sets, and a linear representation is equivalent to a functor to Vect, the category of vector spaces over a field.In this setting, the category of linear representations of over is the functor category → Vect''', which has natural transformations as its morphisms.
Properties
The category of linear representations of a group has a monoidal structure given by the tensor product of representations, which is an important ingredient in Tannaka-Krein duality.Maschke's theorem states that when the characteristic of doesn't divide the order of, the category of representations of over is semisimple.
Restriction and induction
Given a group with a subgroup, there are two fundamental functors between the categories of representations of and : one is a forgetful functor called the restriction functorand the other, the induction functor
When and are finite groups, they are adjoint to each other
a theorem called Frobenius reciprocity.
The basic question is whether the decomposition into irreducible representations behaves under restriction or induction. The question may be attacked for instance by the Mackey theory.