weightNuHighestWeightVectorsInVtensorW -- computes the highest weight vectors of weight nu in $V \otimes W$

Usage:

weightNuHighestWeightVectorsInVtensorW(nu,rhoV,rhoW)
Inputs:
- nu, a list,
- rhoV, an instance of the type LieAlgebraRepresentation,
- rhoW, an instance of the type LieAlgebraRepresentation,
Outputs:
- L, a list,

Description

A highest weight vector is one that is killed by all the raising operators. The more general function weightMuHighestWeightVectorsInW can compute highest weight vectors for any representation by computing the intersection of the kernels of the raising operators restricted to the weight $\mu$ subspace. However, for large representations, this strategy will be slow. In the special case that W is of the form $V \otimesW$, we want a strategy that allows us to work in the weight mu space of $V \otimes W$ without fully computing $V \otimes W$.

Here is our alternative approach. A highest weight vector $v_{\mu}$ of weight $\mu$ will generate an irreducible submodule $V(\mu) \subset V \otimes W$. Therefore, the Casimir operator will act on $V(\mu)$ by a known scalar $c(\mu)$; see casimirScalar. Moreover, the spectrum of the Casimir operator is known. Thus, we can find the weight $\mu$ vectors with eigenvalue $c(\mu)$ by starting with a basis of $V \otimes W$ and iteratively projecting away the components that correspond to the other Casimir scalars.

Let $V$ be the irreducible representation of $sl_4$ with highest weight $\omega_1 + \omega_2$, let $W$ be the irreducible representation with highest weight $\omega_2+\omega_3$, and let $U$ be the irreducible representation with highest weight $\omega_1 + \omega_3$. The multiplicity of $U$ in $V \otimes W$ is 2. In the example below, we compute two highest weight vectors of weight $(1,0,1)$ in $V \otimes W$. We work with the Gelfand-Tsetlin basis in these calculations.

i1 : sl4 = simpleLieAlgebra("A",3);

i2 : LAB=lieAlgebraBasis(sl4);

i3 : V=irreducibleLieAlgebraModule({1,1,0},sl4);

i4 : L = GTrepresentationMatrices(V);

i5 : rhoV = lieAlgebraRepresentation(V,LAB,L);

i6 : W=irreducibleLieAlgebraModule({0,1,1},sl4);

i7 : L = GTrepresentationMatrices(W);

i8 : rhoW = lieAlgebraRepresentation(W,LAB,L);

i9 : weightNuHighestWeightVectorsInVtensorW({1,0,1},rhoV,rhoW)
Constructing the Casimir operator...
Other EVs: {24, 20, 16, 12, 0}
Beginning projections...
    j=0:
        EV 24 complete
        EV 20 complete
        EV 16 complete
        EV 12 complete
        EV 0 complete
    #hwvs=0
    j=1:
        EV 24 complete
        EV 20 complete
        EV 16 complete
        EV 12 complete
        EV 0 complete
    #hwvs=1

                                                                        
o9 = {111A B  - 9A B   - 222A B  - 222A B  + 18A B   + 111A B  - 9A B  +
          0 7     0 11       1 6       2 5      2 10       3 4     3 9  
     ------------------------------------------------------------------------
                                                                      
     111A B  + 3A B  - 148A B  + 84A B  - 42A B  + 14A  B  - 30A  B  +
         5 4     5 9       6 3      8 2      9 1      11 0      14 1  
     ------------------------------------------------------------------------
              768        1536        768       256       2304       1152    
     30A  B , ---A B   - ----A B   + ---A B  - ---A B  + ----A B  - ----A B 
        15 0   37 0 11    37  2 10    37 3 9    37 5 9    37  8 2    37  9 1
     ------------------------------------------------------------------------
       384        192        192
     + ---A  B  + ---A  B  - ---A  B }
        37 11 0    37 14 1    37 15 0

o9 : List

Ways to use weightNuHighestWeightVectorsInVtensorW:

weightNuHighestWeightVectorsInVtensorW(List,LieAlgebraRepresentation,LieAlgebraRepresentation)

For the programmer

The object weightNuHighestWeightVectorsInVtensorW is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2911:0.