Clebsch-Gordan coefficents necessary and sufficent condition to be non-zero I know that the Clebsch-Gordan coefficient, $$\left<J_1, m_1, J_2, m_2|J,M\right>,$$ is zero if the following conditions are not satisfied:
$$|J_1-J_2| \le J \le J_1+J_2,$$
$$m_1+m_2=M,$$
$$|M| \le J.$$
My question is whether it is possible that a Clebsch-Gordan coefficient satisfies these conditions but is still zero? I.e. are these conditions been satisfied a necessary  and sufficient condition for a non-zero Clebsch-Gordan coefficient or only a necessary? and can it be proved either way?
 A: A Clebsch-Gordan coefficient can be zero even if those conditions are satisfied. For example*, using the notations $|J, M\rangle$ and $|J_1, m_1, \, J_2, m_2 \rangle$:
$$
|\,2, 0\,\rangle= \frac{1}{\sqrt{2}}|\,2,+1,\,1,-1\,\rangle -  \frac{1}{\sqrt{2}}|\,2,-1,\,1,+1\,\rangle \,,
$$
so that there is no projection on the $m_1=m_2=0$ state:
$$
\langle J, M |J_1, m_1,\, J_2, m_2 \rangle
= 
\langle \,2,0\,| \,2,0,\,1,0\,   \rangle = 0\,.
$$

*I basically searched for a zero entry in Table 4.7 of David J. Griffiths' Introduction to Quantum Mechanics, 2nd edition.

A: The topic of "accidental" zeros of Clebsh-Gordan coefficients is still active. See this paper as an example of efforts to classify these non-trivial zeros.
A: The example given above of the Clebsch-Gordan $\langle 2,0|2,0,1,0\rangle$ is not an accidental or non-trivial zero! It is a result of the known selection rules. It corresponds to the 3-$j$ case (Mathematica notation)
ThreeJSymbol[{2, 0}, {1, 0}, {2, 0}]

and is zero because the sum of the top 3 three $j$'s, i.e. $2+1+2$, is not even as required (see formula for evaluation of ThreeJSymbol[{j1, 0}, {j2, 0}, {j3, 0}] in any book on 3-$j$ symbols).
A real example of an accidental zero is the case 
ThreeJSymbol[{3, 2}, {3, -2}, {2, 0}] = 0

which is seen to obey the $j_1+j_2+j_3 = \mathrm{even}$ ($3+3+2=8$) selection rule, but is still zero!
There seems to be an infinity of such accidental zeros - see reference below, i.e. paper by Heim et al 1992.
