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Subsections
The functions described in this section are applicable only to almost
reductive structure constant Lie algebras.
The one-dimensional trivial representation of the Lie
algebra L over its base ring.
The standard representation of the (almost) reductive Lie algebra L over its
base ring.
This is the smallest dimensional faithful
representation of L (with a few small exceptions).
The Killing form of L must be nondegenerate.
The adjoint representation of the Lie algebra L acting on itself.
> 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 always 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 behind this function is described in [dG01].
> L:= SimpleLieAlgebra("G", 2, RationalField());
> DimensionOfHighestWeightModule(RootDatum(L), [1,0]);
7
> rho:= HighestWeightRepresentation(L, [1,0]);
> rho(L.1+L.7);
[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]
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 behind this function is described in [dG01].
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]
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