LieAlgebraRepresentation ** LieAlgebraRepresentation -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$

Operator: **
Usage:

V**W
Inputs:
- V, an instance of the type LieAlgebraRepresentation,
- W, an instance of the type LieAlgebraRepresentation,
Outputs:
- U, an instance of the type LieAlgebraRepresentation,

Description

Let $V$ and $W$ be LieAlgebraRepresentations. Then this function computes the action of $\mathfrak{g}$ on $U = V \otimes V$.

In the example below, we compute $V \otimes W$, where $V$ is the adjoint representation of $sl_3$, and $W$ is the standard representation of $sl_3$.

i1 : V = adjointRepresentation("A",2);

i2 : W = standardRepresentation("A",2);

i3 : U = V**W;

i4 : first(U#"RepresentationMatrices")

o4 = | 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 -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  |
     | 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 -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 0 0  0 3 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 2 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 0 0 0 -2 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  2 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  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 -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  -3 0  0 0 0 0 0  0  |
     | 0 0  0 0 0  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 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 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 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 0 0 0 -2 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 |

              24       24
o4 : Matrix QQ   <-- QQ

Ways to use this method:

LieAlgebraRepresentation ** LieAlgebraRepresentation -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$

The source of this document is in LieAlgebraRepresentations/documentation.m2:2545:0.