Pensieve header: Verifying that the representation ρ is Hermitian on u-primitives.

n=6;
DD=DiagonalMatrix[Table[(Subscript[b, i]),{i,n}]];
Unprotect[ConjugateTranspose]; (
ConjugateTranspose /: (M_)\[ConjugateTranspose]/; MatrixQ[M]:= Transpose[M] /. {(Subscript[b, i_]):>-(Subscript[b, i])};
); Protect[ConjugateTranspose];
ρ[x_Plus] := ρ /@ x;
ρ[bs_. a[j_,k_]] := bs*(Table[0, {n},{n}] // ReplacePart[{
{j,k} -> -(Subscript[b, k]), {k,k}->(Subscript[b, j])
}]);
ρ[bs_. t[j_,k_]] := ρ[bs*a[j,k]+bs*a[k,j]];
ρ[bs_. Y[i_,j_,k_]] := ρ[Expand[bs*(
(Subscript[b, i])a[j,k]+(Subscript[b, j])a[k,i]+(Subscript[b, k])a[i,j]-(Subscript[b, j])a[i,k]-(Subscript[b, k])a[j,i]-(Subscript[b, i])a[k,j]
)]];
ρ[bs_. H[i_,j_,k_,l_]] := ρ[Expand[bs*(
(Subscript[b, i])(Subscript[b, k])a[j,l]-(Subscript[b, i])(Subscript[b, l])a[j,k]-(Subscript[b, j])(Subscript[b, k])a[i,l]+(Subscript[b, j])(Subscript[b, l])a[i,k]+(Subscript[b, i])(Subscript[b, k])a[l,j]-(Subscript[b, i])(Subscript[b, l])a[k,j]-(Subscript[b, j])(Subscript[b, k])a[l,i]+(Subscript[b, j])(Subscript[b, l])a[k,i]
)]];

MatrixForm /@ {ρ[7t[2,3]], ρ[7t[2,3]]\[ConjugateTranspose], DD}

MatrixForm /@ {A=ρ[t[2,4]],A\[ConjugateTranspose].DD+DD.A}

MatrixForm /@ {A=ρ[Y[2,3,4]],A\[ConjugateTranspose].DD+DD.A}

MatrixForm /@ {A=ρ[H[2,3,4,5]],A\[ConjugateTranspose].DD+DD.A}

MatrixForm /@ {A=ρ[H[2,3,3,4]],A\[ConjugateTranspose].DD+DD.A}

MatrixForm /@ {A=ρ[H[2,3,2,3]],A\[ConjugateTranspose].DD+DD.A}