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

Class XXIII: (31 11)

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

Eigenvalues[V]

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

eps2,eps3,eps4 have same block size, so we may assume that eps2 <= eps3 <= eps4.

Eigenvalues[W]

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}]
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}]

2 sign possibilities: --++, +--+

pos1={eps1->-1,eps2->-1,eps3->1,eps4->1};
pos2={eps1->1,eps2->-1,eps3->-1,eps4->1};

Eigenvalues[W/.pos1]
Eigenvalues[W/.pos2]

For both configurations, eps2=-1, eps4=1 and eps3 = -eps1

W=W/.{eps2->-1,eps4->1,eps3->-eps1};

W//MatrixForm

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

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/ClassXXIII.pdf"]