Representations

This section describes basic functionality for Lie algebra representations: see Chapter Representations of Lie Groups and Algebras 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 [CMT04].

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 [CMT04].

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 [CMT04].

Example: Standard Representation of a Group of Lie Type (ex-be5644)

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> 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]

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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.

Representations#

Representations