# 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:

```magma
> 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)

```