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  "body": "\n## Definition\n\nLet $V_j$ be the irreducible representation of [[SU(2)|E5P8AG6CK7]] of dimension $2j + 1$, and let $j_1, j_2, j_3, J$ be spins.\nCoupling uses the Clebsch–Gordan coefficients of the Condon–Shortley convention, as in the definition of the [[3-j symbol|YJRJVZ998D]].\nIn the [[tensor product|0BF2DE2SYK]] $V_{j_1} \\otimes V_{j_2} \\otimes V_{j_3}$, let $\\lvert (j_1 j_2) j_{12}, j_3; J M \\rangle$ be the orthonormal vectors obtained by coupling $j_1$ and $j_2$ to $j_{12}$, then $j_{12}$ and $j_3$ to $J$.\nLet $\\lvert j_1, (j_2 j_3) j_{23}; J M \\rangle$ be obtained by coupling $j_2$ and $j_3$ to $j_{23}$ and then $j_1$ and $j_{23}$ to $J$.\nThe *6-j symbol* is the number $\\begin{Bmatrix} j_1 \u0026 j_2 \u0026 j_{12} \\\\ j_3 \u0026 J \u0026 j_{23} \\end{Bmatrix}$ such that\n$$\\langle (j_1 j_2) j_{12}, j_3; J M \\mid j_1, (j_2 j_3) j_{23}; J M \\rangle = (-1)^{j_1 + j_2 + j_3 + J} \\sqrt{(2 j_{12} + 1)(2 j_{23} + 1)}\\, \\begin{Bmatrix} j_1 \u0026 j_2 \u0026 j_{12} \\\\ j_3 \u0026 J \u0026 j_{23} \\end{Bmatrix}.$$\nThe left side does not depend on $M$.\nThe symbol is set to $0$ when one of the couplings is impossible.\n\n## Remarks\n\nA symbol $\\begin{Bmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ j_4 \u0026 j_5 \u0026 j_6 \\end{Bmatrix}$ 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.\nIt is unchanged by any permutation of its columns, and by exchanging the upper and lower entries in any two columns.\n\nExample: take $j_3 = 0$, so $J = j_{12} = c$ and $j_{23} = j_2 = b$, with $j_1 = a$.\nCoupling with the trivial representation changes nothing, so both vectors equal $\\lvert (a b) c; c M \\rangle$ and the overlap is $1$.\nHence $\\begin{Bmatrix} a \u0026 b \u0026 c \\\\ 0 \u0026 c \u0026 b \\end{Bmatrix} = \\frac{(-1)^{a + b + c}}{\\sqrt{(2b+1)(2c+1)}}$.\n\nNon-example: $\\begin{Bmatrix} 1 \u0026 1 \u0026 3 \\\\ 1 \u0026 1 \u0026 1 \\end{Bmatrix} = 0$, because the triad $(1, 1, 3)$ fails the triangle inequality.\n\nThe Racah W-coefficient is the same number with another sign and order: $\\begin{Bmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ j_4 \u0026 j_5 \u0026 j_6 \\end{Bmatrix} = (-1)^{j_1 + j_2 + j_4 + j_5} W(j_1 j_2 j_5 j_4; j_3 j_6)$.\n",
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