Constructing Weight Multisets

In this section, we describe how to construct weight multisets.

TrivialLieRepresentationDecomposition(R) : RootDtm -> LieRepDec
LieRepresentationDecomposition(R) : RootDtm -> LieRepDec
The decomposition multiset of the trivial representation. The root datum R must be weakly simply connected.
LieRepresentationDecomposition(R, v) : RootDtm, ModTupRngElt -> LieRepDec
LieRepresentationDecomposition(R, v) : RootDtm, SeqEnum -> LieRepDec
The decomposition multiset of the highest weight representation with weight v, i.e., the singleton multiset. The root datum R must be weakly simply connected. The weight v must be a sequence of length d or an element of Zd, where d is the dimension of the root datum R.
LieRepresentationDecomposition(R, Wt, Mp) : RootDtm, SeqEnum, SeqEnum -> LieRepDec
The decomposition multiset with weights given by the sequence Wt and multiplicities given by of the sequence Mp. The root datum R must be weakly simply connected. The weights must be a sequences of length d or elements of Zd, where d is the dimension of the root datum R.
AdjointRepresentationDecomposition(R) : RootDtm -> LieRepDec
The decomposition multiset of the adjoint representation. This has the highest root of R as its highest weight with multiplicity one. The root datum R must be weakly simply connected.

Example LieReps_AdjointRepresentation (H111E1)

The adjoint representation: 

> R := RootDatum("D4" : Isogeny := "SC");
> D := AdjointRepresentationDecomposition(R);
> D:Maximal;
Highest weight decomposition of representation of:
     R: Simply connected root datum of dimension 4 of type D4
     Dimension of weight space:4
     Weights:
          [
               (0 1 0 0)
          ]
     Multiplicities:
          [ 1 ]
> HighestRoot(R : Basis := "Weight");
(0 1 0 0)
V2.28, 13 July 2023