GTrepresentationMatrices -- creates a list of matrices for the action of $\mathfrak{g}$ on Gelfand-Tsetlin basis

Usage:

GTrepresentationMatrices(V,lambda)
Inputs:
- V, an instance of the type LieAlgebraModule,
Outputs:
- L, a list,

Description

Currently only supported for $\mathfrak{g} = sl_n$.

Let $\rho: sl_n \rightarrow \mathfrak{gl}(V)$ be a representation where $V$ is irreducible of highest weight $\lambda$. Let $\{B_i\}$ be a basis for $sl_n$. This function creates the list of matrices $\{M_i\}$, where $M_i$ is the matrix of the endomorphism $\rho(B_i)$ with respect to the Gelfand-Tsetlin basis on $V$. See Molev, "Gelfand-Tsetlin bases for classical Lie algebras", 2018 for additional details about the Gelfand-Tsetlin basis.

The output is a list of matrices that may in turn be used to create a representation with (e.g. with lieAlgebraRepresentation).

In the following example, we compute matrix generators for the adjoint representation of $sl_3$ with respect to the Gelfand-Tsetlin basis of $sl_3$, then use these matrices to create a representation that is isomorphic to, but not equal to, the representation created by adjointRepresentation. Finally, we compute an isomorphism between these two representations.

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

o1 = sl3

o1 : simple LieAlgebra

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

o2 = V

o2 : irreducible LieAlgebraModule over sl3

i3 : L=GTrepresentationMatrices(V)

o3 = {| 1 0  0 0 0  0 0 0  |, | 1 0 0  0 0 0 0  0  |, | 0 1 0 0 0 0 0 0 |, |
      | 0 -1 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 -1 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 -2 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 1 0  |  | 0 0 0  0 0 0 -2 0  |  | 0 0 0 0 0 0 0 1 |  |
      | 0 0  0 0 0  0 0 -1 |  | 0 0 0  0 0 0 0  -1 |  | 0 0 0 0 0 0 0 0 |  |
     ------------------------------------------------------------------------
     0 0 1 0 0 0 0   0   |, | 0 0 0 -1 0  3 0 0    |, | 0 0 0 0 0 0 0 0 |, |
     0 0 0 1 0 3 0   0   |  | 0 0 0 0  -2 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 3 0    |  | 0 0 0 0 0 0 0 0 |  |
     0 0 0 0 0 0 3/2 0   |  | 0 0 0 0  0  0 0 3/2  |  | 0 0 1 0 0 0 0 0 |  |
     0 0 0 0 0 0 0   3/2 |  | 0 0 0 0  0  0 0 0    |  | 0 0 0 1 0 0 0 0 |  |
     0 0 0 0 0 0 3/2 0   |  | 0 0 0 0  0  0 0 -3/2 |  | 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 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 1/2 0 0   0   0 0 0 |  | -1/2 0    0   0   0 0  0 0 |
     0 0   0 0   0   0 0 0 |  | 0    -1/2 0   0   0 0  0 0 |
     0 1/2 0 0   0   0 0 0 |  | 1/2  0    0   0   0 0  0 0 |
     0 0   0 1/3 0   1 0 0 |  | 0    0    1/3 0   0 0  0 0 |
     0 0   0 0   2/3 0 0 0 |  | 0    0    0   1/3 0 -1 0 0 |

o3 : List

i4 : LAB=lieAlgebraBasis(sl3);

i5 : rho1=lieAlgebraRepresentation(V,LAB,L)

o5 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl3                                                                                                                                                                                                    }
                              "Module" => V
                              "RepresentationMatrices" => {| 1 0  0 0 0  0 0 0  |, | 1 0 0  0 0 0 0  0  |, | 0 1 0 0 0 0 0 0 |, | 0 0 1 0 0 0 0   0   |, | 0 0 0 -1 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 -1 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 3 0   0   |  | 0 0 0 0  -2 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 -1 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 3 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 2 0 0 0 |  | 0 0 0 0 0 0 3/2 0   |  | 0 0 0 0  0  0 0 3/2  |  | 0 0 1 0 0 0 0 0 |  | 0 1/2 0 0   0   0 0 0 |  | -1/2 0    0   0   0 0  0 0 |
                                                           | 0 0  0 0 -2 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   3/2 |  | 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    -1/2 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 3/2 0   |  | 0 0 0 0  0  0 0 -3/2 |  | 0 0 0 0 0 0 0 0 |  | 0 1/2 0 0   0   0 0 0 |  | 1/2  0    0   0   0 0  0 0 |
                                                           | 0 0  0 0 0  0 1 0  |  | 0 0 0  0 0 0 -2 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 1/3 0   1 0 0 |  | 0    0    1/3 0   0 0  0 0 |
                                                           | 0 0  0 0 0  0 0 -1 |  | 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   2/3 0 0 0 |  | 0    0    0   1/3 0 -1 0 0 |

o5 : LieAlgebraRepresentation

i6 : rho2=adjointRepresentation(sl3)

o6 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl3                                                                                                                                                                                         }
                              "Module" => V
                              "RepresentationMatrices" => {| 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  -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 0 0 0  |  | 0 0  0  0 0 0 1 0 |  | 0  0  0 0 0 0  0 1 |  | 0 0  0  0 0 0 0  0 |  | 0  0 0 -1 0  0 0 0 |  | 0 0 0 0  -1 0 0 0 |
                                                           | 0 0 2 0  0 0  0 0  |  | 0 0 -1 0 0 0 0  0  |  | -2 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  -1 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 0  0  |  | 0  0 0 0 0 0 0 0  |  | 1 -2 0  0 0 0 0 0 |  | 0  0  0 0 0 -1 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  1 0  0 0  |  | 0 0 0  0 1 0 0  0  |  | 0  0 0 1 0 0 0 0  |  | 0 0  -1 0 0 0 0 0 |  | -1 -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 -2 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 |  | 2 -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  1 0  |  | 0 0 0  0 0 0 -2 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 2 0 0  0  0 0 0 |  | 0 0 1 0  0  0 0 0 |
                                                           | 0 0 0 0  0 0  0 -1 |  | 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  1 0 0 |  | 1 1 0 0  0  0 0 0 |

o6 : LieAlgebraRepresentation

i7 : isomorphismOfRepresentations(rho1,rho2)
Length 1 complete. 3 new words found
Length 2 complete. 4 new words found

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

              8       8
o7 : Matrix QQ  <-- QQ

Ways to use GTrepresentationMatrices:

GTrepresentationMatrices(LieAlgebraModule)

For the programmer

The object GTrepresentationMatrices is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2392:0.