momentum operator as well, for . Note that it is common to omit the part. The total angular momentum quantum number must satisfy the triangular condition that such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle. The total number of total angular momentum eigenstates is necessarily equal to the dimension of : As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension , where ranges from to in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to with the two-dimensional representation with . The possible values of are then and . Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation. The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise. The total angular momentum states form an orthonormal basis of : These rules may be iterated to, e.g., combine doublets (=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle), where is the integer floor function; and the number preceding the boldface irreducible representation dimensionality () label indicates multiplicity of that representation in the representation reduction. For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, . Formal definition of Clebsch–Gordan coefficients The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis The expansion coefficients are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as . Another common notation is . Applying the operators to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when Recursion relations The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941. Applying the total angular momentum raising and lowering operators to the left hand side of the defining equation gives Applying the same operators to the right hand side gives where was defined in . Combining these results gives recursion relations for the Clebsch–Gordan coefficients: Taking the upper sign with the condition that gives initial recursion relation: In the Condon–Shortley phase convention, one adds the constraint that (and is therefore also