The functions described in this section are applicable only to almost reductive structure constant Lie algebras.
If L has of large dimension, the step that calculates the information needed to compute preimages for these representations can be quite time consuming. So if there is no requirement for preimages, this step may be skipped by setting the optional argument ComputePreImage to false.
The one-dimensional trivial representation of the Lie algebra L over its base ring.
ComputePreImage: BoolElt Default: true
The adjoint representation of the Lie algebra L acting on itself.
ComputePreImage: BoolElt Default: true
The standard representation of the semisimple Lie algebra L over its base ring. This is the smallest dimensional faithful representation of G (with a few small exceptions). The Killing form of L must be nondegenerate.
> R := RootDatum("A2");
> #CoisogenyGroup(R);
3
> L := LieAlgebra(R, GF(2));
> h := StandardRepresentation(L);
> h(L.1);
[0 0 1]
[0 0 0]
[0 0 0]
> L := LieAlgebra(R, GF(3));
> h := StandardRepresentation(L);
>> h := StandardRepresentation(L);
^
Runtime error in 'StandardRepresentation': Cannot compute the standard
representation in characteristic 3
The coisogeny group of a simply connected root datum always has order one,
so we can compute the standard representation in this case.
> R := RootDatum("A2" : Isogeny:="SC");
> L := LieAlgebra(R, GF(3));
> h := StandardRepresentation(L);
The representation of the Lie algebra L with highest weight w (given either as a vector or as a sequence representing a vector). The result is a function, which for an element of L gives the corresponding matrix. The algorithm used is described in [dG01].
> L:= LieAlgebra("G2", RationalField());
> WeylDimension(RootDatum(L), [1,0]);
7
> rho:= HighestWeightRepresentation(L, [1,0]);
> e, f, h := ChevalleyBasis(L);
> rho(e[1]+f[1]);
[ 0 1 0 0 0 0 0]
[ 1 0 0 0 0 0 0]
[ 0 0 0 -2 0 0 0]
[ 0 0 -1 0 -1 0 0]
[ 0 0 0 -2 0 0 0]
[ 0 0 0 0 0 0 1]
[ 0 0 0 0 0 1 0]
> Codomain(rho);
Full Matrix Lie Algebra of degree 7 over Rational Field
> N := sub<Codomain(rho) | [ rho(x) : x in e ]>;
> Dimension(N);
6
> IsSolvable(N);
true
Given a semisimple Lie algebra L corresponding to a root datum of rank r and a sequence w of non-negative integers of length r, this returns the irreducible L-module with highest weight w. The object returned is a left module over L. The algorithm used is described in [dG01].
Given a sequence Q of left-modules over a Lie algebra, this function returns the module M that is the tensor product of the elements of Q. It also returns a map φ from the Cartesian product P of the modules in Q to M as the second return value. If t is a tuple whose i-th component is an element from the i-th module in Q then φ maps t to the element of M that corresponds to the tensor product of the elements of t.
Given a left-module V over a Lie algebra, and an integer n≥2, this function returns the module M that is the n-th symmetric power of V. It also returns a map f from the n-fold Cartesian product of V to M. This map is multilinear and symmetric, i.e., if two of its arguments are interchanged then the image remains the same. Furthermore, f has the universal property, i.e., any multilinear symmetric map from the n-fold Cartesian product into a vector space W can be written as the composition of f with a map from M into W.
Given a left-module V over a Lie algebra, and an integer 2≤n ≤dim(V), this function returns the module M that is the n-th exterior power of V. It also returns a map f from the n-fold Cartesian product of V to M. This map is multilinear and antisymmetric, i.e., if two of its arguments are interchanged then the image is multiplied by -1. Furthermore, f has the universal property, i.e., any multilinear antisymmetric map from the n-fold Cartesian product into a vector space W can be written as the composition of f with a map from M into W.
> L:= LieAlgebra("G2", Rationals());
> V1:= HighestWeightModule(L, [1,0]);
> V2:= HighestWeightModule(L, [0,1]);
> T,f:= TensorProduct([V1,V2]);
> HighestWeightsAndVectors(T);
[
(1 1),
(2 0),
(1 0)
]
[
[
T: (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
],
[
T: (0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 -3 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
],
[
T: (0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 0
0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
]
]
> DecomposeTensorProduct(RootDatum(L), [1,0], [0,1]);
[
(1 1),
(2 0),
(1 0)
]
[ 1, 1, 1 ]
> f(<V1.2,V2.3>);
T: (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
So we see that the tensor product T decomposes as a direct sum of three
submodules. This information can also be computed by using
DecomposeTensorProduct. However, in the former case,
the corresponding highest-weight vectors are also given.
> E,h:= ExteriorPower(V1, 3);
> h(<V1.1,V1.3,V1.4>);
E: (0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> h(<V1.1,V1.4,V1.3>);
E: (0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> DecomposeExteriorPower( RootDatum(L), 3, [1,0] );
[
(2 0),
(1 0),
(0 0)
]
[ 1, 1, 1 ]
> HighestWeightsAndVectors(E);
[
(2 0),
(1 0),
(0 0)
]
[
[
E: (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0)
],
[
E: (0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0)
],
[
E: (0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0
0)
]
]
The one-dimensional trivial representation of the group of Lie type G over its base ring.
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].
The adjoint (projective) representation of the group of Lie type G over an extension of its base ring, ie. 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].
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.
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].
> 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]