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

Class VI: (1111  1bar(1) )

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=(({{ lambda1, 0, 0, 0, 0, 0}, {0, lambda2, 0, 0, 0, 0}, {0, 0, lambda3, 0, 0, 0}, {0, 0, 0, lambda4, 0, 0}, {0, 0, 0, 0,  sigma1, tau1}, {0, 0, 0, 0, -tau1,  sigma1}}));

Eigenvalues[V]

W=(({{eps1, 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, 0, 1}, {0, 0, 0, 0, 1, 0}}));

Eigenvalues[W]

(* We know that NN should have signature (3,3) and since the first 4 blocks have the same size we shouold have eps1 ≤ eps2 ≤ eps3 ≤ eps4 *)

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

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

(* Permutation 4,4 with leading B *)
S=(({{0, 0, 0, 0, 0, 1}, {(1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0, 0}, {0, (1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0}, {0, 0, 0, 0, 1, 0}, {-(1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0, 0}, {0, -(1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0}}));

Check that S is in 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/ClassVI.pdf"]