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Representations

We provide a simple abstract type hierarchy for representation categories:

abstract type RepresentationCategory <:Category

Representations of Finite groups

Let $G$ be a finite group. We consider the category of finite dimensional $k$-representations of $G$.

GroupRepresentationCategory <: RepresentationCategory

Build it with the constructor

A group representation is defined by a group homomorphism from $G$ into a finite dimensional vector space $k^n$. These objects are of type

GroupRepresentationObject <: RepresentationObject

They are constructed in one of two ways, either by images of generators or by a function

where in both cases the images are required to be fitting MatrixElem objects.

Since group representation categories are tensor categories, we again have methods for the important operations

AbstractAlgebra.direct_sum — Method
direct_sum(ρ::GroupRepresentation, τ::GroupRepresentation, morphisms::Bool = false)

Return the direct sum of representations with the corresponding injections und projections.

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AbstractAlgebra.direct_sum — Method
direct_sum(f::GroupRepresentationMorphism, g::GroupRepresentationMorphism)

Direct sum of morphisms of representations.

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Base.one — Method
one(Rep::GroupRepresentationCategory)

Return the trivial representation.

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Base.parent — Method
parent(ρ::GroupRepresentation)

Return the parent representation category of ρ.

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Base.zero — Method
zero(Rep::GroupRepresentationCategory)

Return the zero reprensentation.

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Hecke.decompose — Method
decompose(σ::GroupRepresentation)

Decompose the representation into a direct sum of simple objects. Return a list of tuples with simple objects and multiplicities.

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Hecke.id — Method
id(ρ::GroupRepresentation)

Return the identity on ρ.

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Hecke.tensor_product — Method
tensor_product(ρ::GroupRepresentation, τ::GroupRepresentation)

Return the tensor product of representations.

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Hecke.tensor_product — Method
tensor_product(f::GroupRepresentationMorphism, g::GroupRepresentationMorphism)

Return the tensor product of morphisms of representations.

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Oscar.morphism — Method
morphism(ρ::GroupRepresentation, τ::GroupRepresentation, m::MatElem; check = true)

Morphism between representations defined by $m$. If check == false equivariancy will not be checked.

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TensorCategories.Hom — Method
Hom(σ::GroupRepresentation, τ::GroupRepresentation)

Return the hom-space of the representations as a vector space.

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