Inducirana upodobitev D nad polinomi v R

skal[f_, g_] := 
      Integrate[
        f g (R^2) Sin[th], {R, 0, 1}, {th, 0, Pi}, {ph, 0, 
          2  Pi}]
r = {R Sin[th] Cos[ph], R Sin[th] Sin[ph], 
          R Cos[th]};

ψ[{x_, y_, z_}, i_] := 
      Part[{(Sqrt[(63/(4  π))]) (1/2) ((x^2) + 
                (y^2) - (2/3) 
                  (z^2)), (Sqrt[(105/(4  π))]) (1/2) ((x^2) - 
                (y^2)), (Sqrt[(35/(4  π))]) 
            (z^2), (Sqrt[(105/(4  π))]) 
            x y, -(Sqrt[(105/(4  π))]) 
            x z, (Sqrt[(105/(4  π))]) y z}, i]

ψ[{x_, y_, z_}, i_] := 
      Part[{(Sqrt[(63/(4  π))]) (1/2) ((x^2) + 
                (y^2) - (2/3) 
                  (z^2)), (Sqrt[(105/(4  π))]) (1/2) ((x^2) - 
                (y^2)), (Sqrt[(105/(4  π))]) 
            x y, (Sqrt[(35/(4  π))]) 
            (z^2), -(Sqrt[(105/(4  π))]) 
            x z, (Sqrt[(105/(4  π))]) y z}, i]

ψ[{x, y, z}, 5]
ψ[r, 1]

MatrixForm[
      Table[skal[ψ[r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]

Definiramo rotacijske matrike za vektorje ex, ey, ez...Vzamemo grupo D3[r]

De=(({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}));D1=(({{-(1/2), ((Sqrt[3])/2), 0}, {-((Sqrt[3])/2), -(1/2), 0}, {0, 0, 1}}));D2=(({{-(1/2), -((Sqrt[3])/2), 0}, {((Sqrt[3])/2), -(1/2), 0}, {0, 0, 1}}));
D3=(({{-1, 0, 0}, {0, 1, 0}, {0, 0, -1}}));D4=(({{(1/2), -((Sqrt[3])/2), 0}, {-((Sqrt[3])/2), -(1/2), 0}, {0, 0, -1}}));D5=(({{(1/2), ((Sqrt[3])/2), 0}, {((Sqrt[3])/2), -(1/2), 0}, {0, 0, -1}}));

D1 . {ex, ey, ez}
D1 . r

MatrixForm[
      Table[skal[ψ[De . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]
MatrixForm[
      Table[skal[ψ[D2 . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]
MatrixForm[
      Table[skal[ψ[D1 . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]
MatrixForm[
      Table[skal[ψ[D3 . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]
MatrixForm[
      Table[skal[ψ[D4 . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]
MatrixForm[
      Table[skal[ψ[D5 . r, i], ψ[r, j]], {i, 1, 6}, {j, 1, 
          6}]]

FullSimplify[
      Table[ψ[De . {x, y, z}, i], {i, 1, 6}]]
FullSimplify[
      Table[ψ[D2 . {x, y, z}, i], {i, 1, 6}]]
FullSimplify[
      Table[ψ[D1 . {x, y, z}, i], {i, 1, 6}]]
FullSimplify[
      Table[ψ[D3 . {x, y, z}, i], {i, 1, 6}]]
FullSimplify[
      Table[ψ[D4 . {x, y, z}, i], {i, 1, 6}]]
FullSimplify[Table[ψ[D5 . {x, y, z}, i], {i, 1, 6}]]