The total number of total angular momentum eigenstates is necessarily equal to the dimension of V 3 : ∑ ∑ J = | j 1 − − j 2 | j 1 + j 2 ( 2 J + 1 ) = ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) . As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension 2 J + 1 , where J ranges from | j 1 − − j 2 | to j 1 + j 2 in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to j 1 = 1 with the two-dimensional representation with j 2 = 1 / 2 . The possible values of J are then J = 1 / 2 and J = 3 / 2 . Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation.