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 \({\mathbb{Z}}^d\), 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 \({\mathbb{Z}}^d\), 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: Adjoint Representation (ex-e0259e)

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)

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