6-j symbol

Remarks

A symbol $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}$ can be nonzero only if each of the triads $(j_1, j_2, j_3)$, $(j_1, j_5, j_6)$, $(j_4, j_2, j_6)$, $(j_4, j_5, j_3)$ satisfies the triangle inequality and has an integer sum. It is unchanged by any permutation of its columns, and by exchanging the upper and lower entries in any two columns.

Example: take $j_3 = 0$, so $J = j_{12} = c$ and $j_{23} = j_2 = b$, with $j_1 = a$. Coupling with the trivial representation changes nothing, so both vectors equal $\lvert (a b) c; c M \rangle$ and the overlap is $1$. Hence $\begin{Bmatrix} a & b & c \\ 0 & c & b \end{Bmatrix} = \frac{(-1)^{a + b + c}}{\sqrt{(2b+1)(2c+1)}}$.

Non-example: $\begin{Bmatrix} 1 & 1 & 3 \\ 1 & 1 & 1 \end{Bmatrix} = 0$, because the triad $(1, 1, 3)$ fails the triangle inequality.

The Racah W-coefficient is the same number with another sign and order: $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1 + j_2 + j_4 + j_5} W(j_1 j_2 j_5 j_4; j_3 j_6)$.