Two-pion isospin decomposition ($1\otimes 1=2\oplus 1\oplus 0$)

Two pions, each of isospin one, have a nine-dimensional charge-state space. Clebsch-Gordan coefficients decompose it into an isospin-two quintet, an isospin-one triplet and an isospin-zero singlet. Under exchange of the two particle slots, the $I=2,0$ states are symmetric and the $I=1$ states are antisymmetric. Because pions have zero spin and obey bosonic exchange symmetry, the $I=1$ two-pion channel requires odd relative orbital angular momentum, while $I=0,2$ require even orbital angular momentum.