dual(LieAlgebraRepresentation) -- creates the dual representation of a Lie algebra representation

i1 : sl6 = simpleLieAlgebra("A",5)

o1 = sl6

o1 : simple LieAlgebra

i2 : Std = standardRepresentation(sl6);

i3 : rho1 = exteriorPower(2,Std);

i4 : rho2 = dual rho1;

i5 : peek(rho1#"Module")

o5 = LieAlgebraModule{cache => CacheTable{}                                  
                      DecompositionIntoIrreducibles => VirtualTally{{0, 1, 0,
                      LieAlgebra => sl6
     ------------------------------------------------------------------------
                }
     0, 0} => 1}

i6 : peek(rho2#"Module")

o6 = LieAlgebraModule{cache => CacheTable{}                                  
                      DecompositionIntoIrreducibles => VirtualTally{{0, 0, 0,
                      LieAlgebra => sl6
     ------------------------------------------------------------------------
                }
     1, 0} => 1}

i7 : M1 = (rho1#"RepresentationMatrices")_6

o7 = | 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 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 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 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 |

              15       15
o7 : Matrix QQ   <-- QQ

i8 : M2 = (rho2#"RepresentationMatrices")_6

o8 = | 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 -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 -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 -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 |

              15       15
o8 : Matrix QQ   <-- QQ

i9 : M2==-transpose(M1)

o9 = true

i10 : rho3 = exteriorPower(4,Std);

i11 : isomorphismOfRepresentations(rho2, rho3)
Length 1 complete. 8 new words found
Length 2 complete. 6 new words found

o11 = | 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 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 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  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  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  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  0 0  0 0 0  0 0  0 0  0 0 0  0 |

               15       15
o11 : Matrix QQ   <-- QQ