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abstract group applied basic symmetry functions basis functions called character coefficients columns components conjugate continuous groups coordinates define the effect definition degeneracy degenerate degenerate energy level denote diagonal dimensional space direct product eigenfunctions energy level equation example follows form a group functions f group multiplication table group of order group operators Group Theory Hamiltonian Hence Hermitian Hermitian conjugate implies inequivalent invariant irreducible subspaces invariant subspaces inverse irreducible invariant space irreducible representations isomorphic labeled lectures linear combination linear operators mapping matrix of dimension matrix representations means Oa operators operator Oa operators which correspond orthogonality relationships P(Aj pji pji pji proof quantum mechanical reducible and irreducible respect rotation group scalar product selection rules set of functions set of matrices set of partners short course similarity transformation space spanned special symmetry functions square matrix statefunctions Suppose symmetry group transform according unit matrix unitary unitary matrix values variables vectors zero