I read that the dimension of the irreducible representations of $SO(3)$ is the dimension of the totally symmetric tensor$S^{i_1i_2...i_j}$ in A.Zee's book. But, in fact, the group's tensor representation can break up into many different inreducible tensors. For example, for a $2$-rank tensor representation: $$ 9=5\oplus3\oplus1 $$ $5$ is the dimension of symmetric traceless tensor, $3$ the antisymmetric tensor, $1$ is the trace.
So, why the dimension of the irreducible representaions of $SO(3)$ is $2j+1$, the dimension of totally symmetric tensor?