UInVtensorW -- computes a basis of a submodule of $V \otimes W$ isomorphic to $U$ with a given highest weight vector

Usage:

UInVtensorW(U,V,W,hwv)
Inputs:
- U, an instance of the type LieAlgebraRepresentation,
- V, an instance of the type LieAlgebraRepresentation,
- W, an instance of the type LieAlgebraRepresentation,
- hwv, a matrix,
Optional inputs:
- SaveAsFunction
Outputs:
- L, a list,

Description

Suppose that an irreducible module $U$ appears in the decomposition of $V \otimes W$ with multiplicity at least one. Then we can find a highest weight vector using weightNuHighestWeightVectorsInVtensorW, and then compute a basis of a submodule in $V \otimes W$ isomorphic to $U$. The basis elements are expressed as polynomials in two sets of variables corresponding to bases of $V$ and $W$, respectively.

Let $V$ be the adjoint representation of $sl_3$, and let $W$ be the standard representation. Then $V \otimes W$ contains a submodule with highest weight $(0,2)$. We compute an explicit basis for this submodule.

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

o1 = sl3

o1 : simple LieAlgebra

i2 : V = adjointRepresentation(sl3);

i3 : W = standardRepresentation(sl3);

i4 : T = V**W;

i5 : hwv = weightMuHighestWeightVectorsInW({0,2},T)

o5 = | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | -1 |
     | 0  |
     | 0  |
     | 0  |
     | 1  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |
     | 0  |

              24       1
o5 : Matrix QQ   <-- QQ

i6 : Umod = irreducibleLieAlgebraModule({0,2},sl3);

i7 : LAB = lieAlgebraBasis(sl3);

i8 : U = lieAlgebraRepresentation(Umod,LAB,GTrepresentationMatrices(Umod));

i9 : L = UInVtensorW(U,V,W,hwv)
Length 1 complete. 2 new words found
Length 2 complete. 3 new words found

o9 = {- A B  + A B , A B  - A B  + A B , A B  + A B  + A B  - A B , - 2A B  +
         3 0    4 1   1 0    2 1    4 2   0 1    1 1    3 2    5 0      2 2  
     ------------------------------------------------------------------------
     2A B , 2A B  + 2A B  - 2A B , 4A B  - 4A B }
       6 0    0 2     6 1     7 0    5 2     7 1

o9 : List

Ways to use UInVtensorW:

UInVtensorW(LieAlgebraRepresentation,LieAlgebraRepresentation,LieAlgebraRepresentation,Matrix)
UInVtensorW(LieAlgebraRepresentation,LieAlgebraRepresentation,LieAlgebraRepresentation,RingElement) (missing documentation)

For the programmer

The object UInVtensorW is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2960:0.