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angular momentum antisymmetric atom axes basis functions character table commutes configuration contains coordinates corresponding cosets crystal field cycles d-electrons define degeneracy degenerate determinant diagonal matrix dimension dimensionality direct product direct sum eigenfunctions eigenvalues electrons example form bases given group elements group of degree group representation group theory Hamiltonian Hence Hermitean matrix identity integral interchanges invariant inverse irreducible representations lattice linear combinations matrix elements molecule multiplication table n-fold rotation normal divisor Note null matrix number of elements octahedral field octahedral group one-dimensional one-electron orbital orthogonality theorem particles permutation group perturbed plane problem properties quantum numbers relation representation based rotation group Schur's Lemma sentation similarity transformation Similarly simply singlet spherical harmonics spin operators splitting strong field subgroup Suppose symmetric group symmetry operations tetragonal group tetrahedral group transpositions triplet unitary matrix wave functions write Xly2 ylX2 yly2 ylZ2 zero Zly2 zlz2