Macaulay2 » Documentation
Packages » LieAlgebraRepresentations :: weightMuHighestWeightVectorsInW
next | previous | forward | backward | up | index | toc

weightMuHighestWeightVectorsInW -- computes the highest weight vectors of weight mu in W

Description

A highest weight vector is one that is killed by all the raising operators. We compute the intersection of the kernels of the raising operators restricted to the weight $\mu$ subspace in $W$.

Let $V$ be the adjoint representation of $sl_3$. Then $V$ has highest weight $(1,1)$ and dimension 8, and the multiplicity of $V$ in $W = V \otimes V$ is 2. In the example below, we compute two highest weight vectors of $(1,1)$ in $V \otimes V$. We work with the Gelfand-Tsetlin basis in these calculations.

i1 : sl3=simpleLieAlgebra("A",2)

o1 = sl3

o1 : simple LieAlgebra
 
i2 : V=irreducibleLieAlgebraModule({1,1},sl3)

o2 = V

o2 : irreducible LieAlgebraModule over sl3
 
i3 : LAB = lieAlgebraBasis("A",2);
 
i4 : L = GTrepresentationMatrices(V);
 
i5 : V=lieAlgebraRepresentation(V,LAB,L);
 
i6 : W = V**V;
 
i7 : weightMuHighestWeightVectorsInW({1,1},W)

o7 = | 0   0    |
     | 0   0    |
     | 0   0    |
     | 1/2 3/2  |
     | 0   0    |
     | 1/2 -1/2 |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | -1  -3   |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | -2  0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 1   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   1    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |
     | 0   0    |

              64       2
o7 : Matrix QQ   <-- QQ

This function works for any representation $W$. There are also specialized functions for the cases where $W$ has the form $\operatorname{Sym}^d W$ or $V \otimes W$.

See also

Ways to use weightMuHighestWeightVectorsInW:

  • weightMuHighestWeightVectorsInW(List,LieAlgebraRepresentation)

For the programmer

The object weightMuHighestWeightVectorsInW is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2721:0.