[GAP Forum] Re: simple Lie algebra questions

[degraaf at science.unitn.it]

Dear R. N. Tsai,

You asked:

> (1) How do you get all the weights of a representation; I found the function
that
>      provides the dominant weights, but I'd like to get an explicit list of
all.
>      The total number of such weights (including multiplicity) would be the
dimension
 >     of the rep and their ordering is hopefully related to the ordering of the
basis
>      vectors of the rep.

It is possible to find the other weights of an irreducible highest weight
representation by taking the orbit of the dominant weights under the
action of the Weyl group. GAP contains a function for computing such an orbit,
called WeylOrbitIterator. For example

gap> L:= SimpleLieAlgebra("A",2,Rationals);
<Lie algebra of dimension 8 over Rationals>
gap> R:= RootSystem( L );
<root system of rank 2>
gap> dc:= DominantCharacter( L, [2,2] );
[ [ [ 2, 2 ], [ 0, 3 ], [ 3, 0 ], [ 1, 1 ], [ 0, 0 ] ], [ 1, 1, 1, 2, 3 ] ]
gap> W:= WeylGroup( R );
Group([ [ [ -1, 1 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 1, -1 ] ] ])
gap> it:= WeylOrbitIterator( W, [0,3] );
<iterator>
gap> while not IsDoneIterator( it ) do Print( NextIterator( it ), "\n" ); od;
[ 0, 3 ]
[ -3, 0 ]
[ 3, -3 ]


this lists all the weights in the orbit of the weight [0,3].

>  (2) "BilinearFormMat" returns a matrix that has the inner product of the
roots <ai,aj>
>     for example :
>     L:=SimpleLieAlgebra("F",4,Rationals);
>     R:=RootSystem(L);
>     C:=CartanMatrix(R);
>     K:=BilinearFormMat(R);
>     PrintArray(C);
>     [ [   2,   0,  -1,   0 ],
>       [   0,   2,   0,  -1 ],
>       [  -1,   0,   2,  -1 ],
>       [   0,  -1,  -2,   2 ] ]
>     cartan matrix looks familar
>     PrintArray(K);
>    [ [   1/18,      0,  -1/36,      0 ],
>      [      0,    1/9,      0,  -1/18 ],
>      [  -1/36,      0,   1/18,  -1/18 ],
>      [      0,  -1/18,  -1/18,    1/9 ] ]
>     this doesn't look too familiar, but
>    PrintArray(18*K);
>   [ [     1,     0,  -1/2,     0 ],
>     [     0,     2,     0,    -1 ],
>     [  -1/2,     0,     1,    -1 ],
>     [     0,    -1,    -1,     2 ] ]
>   with the factor 18 this is a more familiar normalization of the root
>lengths
>   (other algebras have different factors). There's probably a good reason for
>   normalizing the root lengths this way; I'd like to know that if possible.
>

As far as I know there is no particular reason for choosing this
normalisation. The system just returns a matrix of the bilinear form,
then the user can normalise it in the way he/she likes.

Best wishes,

Willem