casimirOperator -- computes the Casimir operator associated to a representation

Usage:

casimirOperator(rho)
Inputs:
- rho, an instance of the type LieAlgebraRepresentation,
Outputs:
- M, a matrix,

Description

Let $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ be a Lie algebra representation.

Let $\{B_i\}$ be a basis of $\mathfrak{g}$, and let $\{B_i^{*}\}$ be the dual basis with respect to the Killing form. The Casimir operator is

$\operatorname{Cas} = \sum_{i} \rho(B_i^*) \rho(B_i)$.

Recall that in creating a LieAlgebraBasis, we compute the dual basis $\{B_i^{*}\}$. This makes it straightforward to compute the Casimir operator.

If $V$ is irreducible with highest weight $\lambda$, then $\operatorname{Cas} = c(\lambda) \operatorname{Id}$, where $c(\lambda)$ is the scalar computed by casimirScalar.

We compute the Casimir operator for the symmetric square of the standard representation of $sl_3$. Since this is an irreducible representation, we get a scalar multiple of the identity.

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

o1 = sl3

o1 : simple LieAlgebra

i2 : V = standardRepresentation(sl3)

o2 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl3                                                                                      }
                              "Module" => LL   (sl3)
                                            1,0
                              "RepresentationMatrices" => {| 1 0  0 |, | 0 0 0  |, | 0 1 0 |, | 0 0 0 |, | 0 0 1 |, | 0 0 0 |, | 0 0 0 |, | 0 0 0 |}
                                                           | 0 -1 0 |  | 0 1 0  |  | 0 0 0 |  | 0 0 1 |  | 0 0 0 |  | 1 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 1 0 |  | 1 0 0 |

o2 : LieAlgebraRepresentation

i3 : S2V=symmetricPower(2,V);

i4 : casimirScalar(S2V#"Module")

     20
o4 = --
      3

o4 : QQ

i5 : CasS2V = casimirOperator(S2V)

o5 = | 20/3 0    0    0    0    0    |
     | 0    20/3 0    0    0    0    |
     | 0    0    20/3 0    0    0    |
     | 0    0    0    20/3 0    0    |
     | 0    0    0    0    20/3 0    |
     | 0    0    0    0    0    20/3 |

              6       6
o5 : Matrix QQ  <-- QQ

Next, we compute the Casimir operator for $\operatorname{Sym^2} \operatorname{Sym^2} \mathbb{C}^3$. It has two distinct eigenvalues. The eigenvalues match the Casimir scalars of the irreducible submodules appearing in the decomposition of $\operatorname{Sym^2} \operatorname{Sym^2} \mathbb{C}^3$, and the multiplicities of the eigenvalues match the dimensions of these submodules.

i6 : S2S2V=symmetricPower(2,S2V);

i7 : CasS2S2V = casimirOperator(S2S2V)

o7 = | 56/3 0    0    0    0    0    0    0    0    0    0    0    0    0   
     | 0    56/3 0    0    0    0    0    0    0    0    0    0    0    0   
     | 0    0    56/3 0    0    0    0    0    0    0    0    0    0    0   
     | 0    0    0    32/3 0    0    4    0    0    0    0    0    0    0   
     | 0    0    0    0    32/3 0    0    4    0    0    0    0    0    0   
     | 0    0    0    0    0    32/3 0    0    0    0    0    4    0    0   
     | 0    0    0    8    0    0    44/3 0    0    0    0    0    0    0   
     | 0    0    0    0    8    0    0    44/3 0    0    0    0    0    0   
     | 0    0    0    0    0    0    0    0    56/3 0    0    0    0    0   
     | 0    0    0    0    0    0    0    0    0    44/3 0    0    8    0   
     | 0    0    0    0    0    0    0    0    0    0    32/3 0    0    4   
     | 0    0    0    0    0    8    0    0    0    0    0    44/3 0    0   
     | 0    0    0    0    0    0    0    0    0    4    0    0    32/3 0   
     | 0    0    0    0    0    0    0    0    0    0    8    0    0    44/3
     | 0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     | 0    0    0    0    0    0    0    0    0    0    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    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    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    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    |
     56/3 0    0    0    0    0    0    |
     0    56/3 0    0    0    0    0    |
     0    0    56/3 0    0    0    0    |
     0    0    0    32/3 4    0    0    |
     0    0    0    8    44/3 0    0    |
     0    0    0    0    0    56/3 0    |
     0    0    0    0    0    0    56/3 |

              21       21
o7 : Matrix QQ   <-- QQ

i8 : tally eigenvalues CasS2S2V

o8 = Tally{6.66667 => 6 }
           18.6667 => 15

o8 : Tally

i9 : peek S2S2V

o9 = LieAlgebraRepresentation{Basis => Enhanced basis of sl3                                                                                                                                                                                                                                                                                                                                                                                                    }
                              Module => LL   (sl3) ++ LL   (sl3)
                                          0,2           4,0
                              RepresentationMatrices => {| 4 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 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 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 1 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 2 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 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 4 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 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 -1 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 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 2 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 4 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 0 1 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 1 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 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 1 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 1 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 2 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 -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 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 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 1 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  2 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 1 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 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 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 1 0 0 2 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 2 0 0 1 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 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 -2 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 0 0 0 4 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 1 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 2 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 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 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 2 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 2 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 2 0 0 |  | 0 0 0 0 2 0 0 1 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 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 -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 2 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 1 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 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  -2 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 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 1 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 -1 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 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 1 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 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  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 2 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 2 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 1 0 0 2 0 0 0 0 0 0 0 0 |  | 0 0 0 0 2 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 1 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 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 4 |  | 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 1 0 0 0 0 0 0 0 |  | 0 0 0 0 0 2 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 -4 0  0  0  0  0 |  | 0 0 0  0 0 0  0 0 0 0 0  0  0 0  0  4 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 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 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 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 2 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 1 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 4 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  -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 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 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 1 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  -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 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 1 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 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  -1 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 4 |  | 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 2 2 0 0 |  | 0 0 0 0 0 0 0 0 0 0 1 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  -4 |  | 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 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 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |

i10 : V40 = irreducibleLieAlgebraModule({4,0},sl3);

i11 : dim V40

o11 = 15

i12 : casimirScalar(V40)

      56
o12 = --
       3

o12 : QQ

i13 : V02 = irreducibleLieAlgebraModule({0,2},sl3);

i14 : dim V02

o14 = 6

i15 : casimirScalar(V02)

      20
o15 = --
       3

o15 : QQ

Ways to use casimirOperator:

casimirOperator(LieAlgebraRepresentation)

For the programmer

The object casimirOperator is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1830:0.