# Representations

This section describes basic functionality for Lie algebra representations: see Chapter [Representations of Lie Groups and Algebras](../RepresentationsOfLieGroupsAndAlgebras/index-representations-of-lie-groups-and-algebras.md#chapliereps) for more functions for highest weight representations and decompositions.

## `StandardRepresentation(G): GrpLie -> Map`

The standard (projective) representation of the semisimple group of Lie type $G$ over an extension its base ring. In other words, the smallest dimension highest-weight representation. For the classical groups, this is the natural representation. If this is a projective representation rather than a linear representation, a warning is given. This is constructed from the corresponding Lie algebra representation, using the algorithm in [[Cohen *et al.*, 2004](../../references.md#cite-cohenmurraytaylor)].

## `AdjointRepresentation(G): GrpLie -> Map, AlgLie`

The adjoint (projective) representation of the group of Lie type $G$ over an extension of its base ring, i.e. the representation given by the action of $G$ on its Lie algebra. The Lie algebra itself is the second returned value. This is constructed from the corresponding Lie algebra representation, using the algorithm in [[Cohen *et al.*, 2004](../../references.md#cite-cohenmurraytaylor)].

## `LieAlgebra(G): GrpLie -> AlgLie, Map`

The Lie algebra of the group of Lie type $G$, together with the adjoint representation. If this is a projective representation rather than a linear representation, a warning is given.

## `HighestWeightRepresentation(G, v): GrpLie, . -> Map`

The highest weight (projective) representation with highest weight $v$ of the group of Lie type $G$ over an extension of its base ring. If this is a projective representation rather than a linear representation, a warning is given. This is constructed from the corresponding Lie algebra representation, using the algorithm in [[Cohen *et al.*, 2004](../../references.md#cite-cohenmurraytaylor)].

## `Example: Standard Representation (ex-be5644)`

```magma
> G := GroupOfLieType("A2", Rationals() : Isogeny := "SC");
> rho := StandardRepresentation(G);
> rho(elt< G | 1 >);
[ 0 -1  0]
[ 1  0  0]
[ 0  0  1]
> rho(elt<G | <2,1/2> >);
[  1   0   0]
[  0   1   0]
[  0 1/2   1]
> rho(elt< G | VectorSpace(Rationals(),2)![3,5] >);
[  3   0   0]
[  0 5/3   0]
[  0   0 1/5]
>
> G := GroupOfLieType("A2", Rationals());
> Invariants(CoisogenyGroup(G));
[ 3 ]
> rho := StandardRepresentation(G);
Warning: Projective representation
> BaseRing(Codomain(rho));
Algebraically closed field with no variables
> rho(elt< G | VectorSpace(Rationals(),2)![3,1] >);
[r1  0  0]
[ 0 r2  0]
[ 0  0 r2]
> rho(elt< G | VectorSpace(Rationals(),2)![3,1] >)^3;
[  9   0   0]
[  0 1/3   0]
[  0   0 1/3]

```

## `ContravariantForm(ρ): Map[GrpLie,GrpMat] -> AlgMatElt`

A contravariant form for the image of the matrix representation $\rho$ of a group of Lie type.

## `GeneralisedRowReduction(ρ): Map -> Map`

## `RowReductionHomomorphism(ρ): Map -> Map`

## `Inverse(ρ): Map -> Map`

Given a projective matrix representation $\rho:G\to {\operatorname{GL}}_m(k)$, return its inverse.