SetDirectory["/www/user/fdahl/papers/Conjugation/"];
<<kappaLib.m
<<Petrov.m

Class XIX: (51)

B=(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}}));

V=(({{ lam1, 1, 0, 0, 0, 0}, {0, lam1, 1, 0, 0, 0}, {0, 0, lam1, 1, 0, 0}, {0, 0, 0, lam1, 1, 0}, {0, 0, 0, 0,  lam1, 0}, {0, 0, 0, 0, 0,  lam2}}));

Eigenvalues[V]

W=(({{0, 0, 0, 0, eps1, 0}, {0, 0, 0, eps1, 0, 0}, {0, 0, eps1, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, eps2}}));

Here we may not assume that eps1 <= eps2.

Eigenvalues[W]

Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->-1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->1}]

We may assume that eps2 = -eps1

W=W/.{eps2->-eps1};
Eigenvalues[W]

(* Found using FinsSPermutations.m *)
S=(({{0, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, (1/(Sqrt[2])), 0, 0, (1/(Sqrt[2]))}, {eps1, 0, 0, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {0, 0, (eps1/(Sqrt[2])), 0, 0, -(eps1/(Sqrt[2]))}}));

Check that S is in the set mathcal(S)

Transpose[S].B.S==W

Compute result

res=S.V.Inverse[S];
r=Simplify[res];
r//MatrixForm

Petrov[r]

Export notebook as .pdf

NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/ClassXIX.pdf"]