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

Class XVIII: (2 1111)

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, 0, 0, 0, 0}, {0, 0, lam2, 0, 0, 0}, {0, 0, 0, lam3, 0, 0}, {0, 0, 0, 0,  lam4, 0}, {0, 0, 0, 0, 0,  lam5}}));

Eigenvalues[V]

W=(({{0, eps1, 0, 0, 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, eps2, 0, 0, 0}, {0, 0, 0, eps3, 0, 0}, {0, 0, 0, 0, eps4, 0}, {0, 0, 0, 0, 0, eps5}}));

Eigenvalues[W]

We may assume that eps2 <= esp3 <= esp4 <= eps5

Wp= W/.eps1->1;
Sort[Eigenvalues[Wp]/.{eps2->-1,eps3->-1,eps4->-1,eps5->-1}]
Sort[Eigenvalues[Wp]/.{eps2->-1,eps3->-1,eps4->-1,eps5->1}]
Sort[Eigenvalues[Wp]/.{eps2->-1,eps3->-1,eps4->1,eps5->1}]
Sort[Eigenvalues[Wp]/.{eps2->-1,eps3->1,eps4->1,eps5->1}]
Sort[Eigenvalues[Wp]/.{eps2->1,eps3->1,eps4->1,eps5->1}]

Wm= W/.eps1->-1;
Sort[Eigenvalues[Wm]/.{eps2->-1,eps3->-1,eps4->-1,eps5->-1}]
Sort[Eigenvalues[Wm]/.{eps2->-1,eps3->-1,eps4->-1,eps5->1}]
Sort[Eigenvalues[Wm]/.{eps2->-1,eps3->-1,eps4->1,eps5->1}]
Sort[Eigenvalues[Wm]/.{eps2->-1,eps3->1,eps4->1,eps5->1}]
Sort[Eigenvalues[Wm]/.{eps2->1,eps3->1,eps4->1,eps5->1}]

We see: eps2, eps3, eps4, eps5 are fixed, but eps1 is arbitrary:

W=W/.{eps2->-1,eps3->-1,eps4->1,eps5->1};
Eigenvalues[W]

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

Check that S is in the set mathcal(S)

Transpose[S].B.S==W

Compute result

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

Petrov[res]

Export notebook as .pdf

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