basisWordsFromMatrixGenerators -- express each basis element of $V(\lambda)$ as a linear combination of words in the lowering operators applied to the highest weight vector

Usage:

basisWordsFromMatrixGenerators(V)
Inputs:
- V, an instance of the type LieAlgebraRepresentation,
Outputs:
- L, a list,

Description

Irreducible Lie algebra modules are cyclic modules. That is, it is possible to write each element of $V(\lambda)$ as a linear combination of words in the lowering operators applied to the highest weight vector. In particular, we can do this for elements of the basis of $V(\lambda)$ that is used to write the matrix generators of the representation $\rho$.

The output may be parsed as follows. Suppose that we order the lowering operators of $\mathfrak{g}$ as $Y_0,\ldots,Y_k$. Then if the output indicates that $v$ is represented by a word with terms {{{1}, 2}, {{0, 2}, 1}}, this means $v = 2 Y_1.v_\lambda + Y_0.Y_2.v_\lambda$, where $v_\lambda$ represents the highest weight vector.

In the example below, we compute the words that yield the Gelfand-Tsetlin basis for the adjoint representation of $sl_3$.

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

i2 : LAB = lieAlgebraBasis("A",2);

i3 : V=irreducibleLieAlgebraModule({1,1},sl3);

i4 : rho=lieAlgebraRepresentation(V,LAB,GTrepresentationMatrices(V));

i5 : basisWordsFromMatrixGenerators(rho)
Length 1 complete. 3 new words found
Length 2 complete. 4 new words found

o5 = {LieAlgebraModuleElement{"LieAlgebraModule" => V},
                              "Terms" => {{{}, 1}}     
                                                       
                                                       
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V},
                             "Terms" => {{{0}, 1}}    
                                                      
                                                      
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V},
                             "Terms" => {{{1}, 1}}    
                                                      
                                                      
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V },
                             "Terms" => {{{0, 1}, 1}}  
                                                       
                                                       
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V  },
                             "Terms" => {{{0, 2}, -2}}  
                                                        
                                                        
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V           },
                             "Terms" => {{{2}, 2}, {{0, 1}, 1}}  
                                                                 
                                                                 
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V },
                             "Terms" => {{{1, 2}, 3}}  
                                                       
                                                       
     ------------------------------------------------------------------------
     LieAlgebraModuleElement{"LieAlgebraModule" => V   }}
                                                    3
                             "Terms" => {{{2, 2}, - -}}
                                                    2

o5 : List

Ways to use basisWordsFromMatrixGenerators:

basisWordsFromMatrixGenerators(LieAlgebraRepresentation)

For the programmer

The object basisWordsFromMatrixGenerators is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1986:0.