constraint that , while is due to the time-reversed nature of . Relation to Wigner D-matrices Relation to spherical harmonics In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics: It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms of a single spherical harmonic: Other Properties Clebsch–Gordan coefficients for specific groups For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the special unitary group SU(n) are known. In particular, SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists that relates the up, down, and strange quarks. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available. Clebsch–Gordan coefficients for symmetric group are also known as Kronecker coefficients. See also 3-j symbol 6-j symbol 9-j symbol Racah W-coefficient Spherical harmonics Spherical basis Tensor products of representations Associated Legendre polynomials Angular momentum Angular momentum coupling Total angular momentum quantum number Azimuthal quantum number Table of Clebsch–Gordan coefficients Wigner D-matrix Wigner–Eckart theorem Angular momentum diagrams (quantum mechanics) Clebsch–Gordan coefficient for SU(3) Littlewood–Richardson coefficient Remarks Notes References Albert Messiah (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. External links Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows Web interface for tabulating SU(N) Clebsch–Gordan coefficients Further reading Rotation in three dimensions Rotational symmetry Representation theory of Lie groups Quantum mechanics Mathematical physics