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# weightMuHighestWeightVectorsInSymdW -- computes the highest weight vectors of weight mu in $\operatorname{Sym}^d W$

- Usage:
    `weightMuHighestWeightVectorsInSymdW(mu,d,rhoW)`
- Inputs:
  - mu, a list,
  - 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 $\operatorname{Sym}^d W$, we want a strategy that allows us to work in the weight mu space of $\operatorname{Sym}^d W$ without fully computing $\operatorname{Sym}^d W$.

Here is our alternative approach. A highest weight vector $v_{\mu}$ of weight $\mu$ will generate an irreducible submodule $V(\mu) \subset \operatorname{Sym}^d W$. 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 monomial basis of $\operatorname{Sym}^d W$ and iteratively projecting away the components that correspond to the other Casimir scalars.

Let $W$ be the irreducible representation of $sl_4$ with highest weight $\omega_1 + \omega_2$, and let $V(2,0,1)$ be the irreducible representation with highest weight $2\omega_1 + \omega_3$. The multiplicity of $V(2,0,1)$ in $\operatorname{Sym}^3 W$ is 2. In the example below, we compute two highest weight vectors of weight $(2,0,1)$ in $\operatorname{Sym}^3 W$. We work with the Gelfand-Tsetlin basis in these calculations.

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

    i2 : LAB=lieAlgebraBasis(sl4);

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

    i4 : L = GTrepresentationMatrices(W);

    i5 : rhoW = lieAlgebraRepresentation(W,LAB,L);

    i6 : weightMuHighestWeightVectorsInSymdW({2,0,1},3,rhoW)
    Constructing the Casimir operator...
    Other EVs: {183/4, 127/4, 111/4, 95/4, 87/4, 71/4, 63/4, 39/4, 15/4}
    Beginning projections...
        j=0:
            EV 183/4 complete
            EV 127/4 complete
            EV 111/4 complete
            EV 95/4 complete
            EV 87/4 complete
            EV 71/4 complete
            EV 63/4 complete
            EV 39/4 complete
            EV 15/4 complete
        #hwvs=0
        j=1:
            EV 183/4 complete
            EV 127/4 complete
            EV 111/4 complete
            EV 95/4 complete
            EV 87/4 complete
            EV 71/4 complete
            EV 63/4 complete
            EV 39/4 complete
            EV 15/4 complete
        #hwvs=1
        j=2:
            EV 183/4 complete
            EV 127/4 complete
            EV 111/4 complete
            EV 95/4 complete
            EV 87/4 complete
        j=3:
            EV 183/4 complete
            EV 127/4 complete
            EV 111/4 complete
            EV 95/4 complete
            EV 87/4 complete
            EV 71/4 complete
            EV 63/4 complete
            EV 39/4 complete
            EV 15/4 complete
        #hwvs=1

               2
    o6 = {- 34B B   + 136B B B   - 82B B B   + 27B B B   + 18B B B   + 9B B B   +
               0 13       0 2 12      0 3 11      0 3 15      0 5 11     0 5 15
         ------------------------------------------------------------------------

         28B B B  - 18B B B   - 136B B B  + 28B B B   - 54B B B   + 80B B B  -
            0 6 9      0 6 14       0 7 8      1 2 11      1 2 15      1 6 8
         ------------------------------------------------------------------------
             2                                                         2
         204B B   + 204B B B  + 204B B B  - 96B B B  + 18B B B   - 204B B  +
             2 10       2 3 9       2 4 8      2 5 9      2 5 14       3 8
         ------------------------------------------------------------------------
                       2      2520          1260          840          420
         96B B B  - 36B B , - ----B B B   - ----B B B   - ---B B B   - ---B B B
            3 5 8      5 8     17  0 3 11    17  0 3 15    17 0 5 11    17 0 5 15
         ------------------------------------------------------------------------
           5040         840          5040          2520          10080
         + ----B B B  + ---B B B   + ----B B B   + ----B B B   - -----B B B  -
            17  0 6 9    17 0 6 14    17  1 2 11    17  1 2 15     17  1 6 8
         ------------------------------------------------------------------------
         5040         840          5040         1680 2
         ----B B B  - ---B B B   + ----B B B  + ----B B }
          17  2 5 9    17 2 5 14    17  3 5 8    17  5 8

    o6 : List

## Ways to use weightMuHighestWeightVectorsInSymdW:

- `weightMuHighestWeightVectorsInSymdW(List,ZZ,LieAlgebraRepresentation)`

## For the programmer

The object weightMuHighestWeightVectorsInSymdW is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:2767:0.