History of Wigner $9j$ symbols

     constraints: $j_9$ is a nonnegative integer or half-integer Comments:-  comment-symmetry: A Wigner $9j$ symbol has 72 row, column and transposition symmetries.+  comment-symmetry: 'A Wigner $9j$ symbol has 72 row, column and transposition symmetries.     Even permutations of rows or columns and transposition leave the symbol unchanged.     An odd row or column permutation multiplies it by $(-1)^R$, where $R=j_1+\cdots+j_9$.     Each class is stored once, under the lexicographically largest array among the-    72 images, with the value for that representative.+    72 images, with the value for that representative, so a symbol in another arrangement+    is found by applying these symmetries first: when $R$ is odd, an odd permutation+    of the rows or of the columns has the opposite sign.'   comment-admissible: 'A Wigner $9j$ symbol is zero unless each of the three row triads     $(j_1,j_2,j_3)$, $(j_4,j_5,j_6)$ and $(j_7,j_8,j_9)$, and each of the three column     triads $(j_1,j_4,j_7)$, $(j_2,j_5,j_8)$ and $(j_3,j_6,j_9)$, satisfies the triangle     inequalities and has an integer sum. This condition is necessary but not sufficient:-    of the 803 nontrivial canonical classes with all $j_i\leq5/2$ satisfying these-    six triad conditions, 39 vanish.'+    of the 803 canonical classes with all $j_i\leq5/2$, other than the all-zero symbol,+    that satisfy these six triad conditions, 39 vanish.'   comment-zeros: Rows whose value is zero are omitted, as is the trivial all-zero     symbol $\begin{Bmatrix}0&0&0\\0&0&0\\0&0&0\end{Bmatrix}$, whose value is $1$.-  comment-algebraic: Every stored value is algebraic. The generator represents each-    term of the finite sums as a rational multiple of the square root of a rational-    number before summing it in ball arithmetic.+  comment-algebraic: Every value stored here is a rational number or a rational multiple+    of the square root of a rational number. Formulas:   formula-6j-sum: The $9j$ symbol is the finite sum $\begin{aligned} \begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\\j_7&j_8&j_9\end{Bmatrix}

  -      wigner_9j(0, 1, 1, 1, 1, 1, 1, 1, 1)    # 1/18'+      wigner_9j(1, 1, 1, 1, 1, 1, 1, 1, 0)    # 1/18+++      # Sage''s wigner_9j raises on many arrays stored here, for example++      # at (1/2, 1/2, 0, 1/2, 1/2, 0, 0, 0, 0).' Similar tables: - table: HREF{T253}[Wigner $6j$ symbols]

   relation: Wigner $9j$ symbols are finite sums of products of six Wigner $3j$ symbols - table: HREF{T35}[Algebraic numbers of degree 2]-  relation: individual summands in the finite sums are rational numbers or quadratic-    algebraic numbers+  relation: the non-rational values stored here are quadratic algebraic numbers Links:   DLMF-9j:

      Before any entry is returned, the generator recomputes every row at all 72 symmetry-    images and requires the row, column and transposition phase rule stated in the-    comments. It checks every row against the finite sum of Wigner $3j$ symbols in-    CITE{formula-3j-sum}, and it checks the zero-entry formula CITE{formula-zero-entry}-    wherever that formula applies.'+    images and requires the row, column and transposition phase rule CITE{comment-symmetry}.+    It checks every row against the finite sum of Wigner $3j$ symbols in CITE{formula-3j-sum},+    and it checks the zero-entry formula CITE{formula-zero-entry} wherever that formula+    applies.' Display properties:   number-header: $\begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\\j_7&j_8&j_9\end{Bmatrix}$