History of Wigner $6j$ symbols

     constraints: $j_6$ is a nonnegative integer or half-integer Comments:-  comment-admissible: The four triads $(j_1,j_2,j_3)$, $(j_1,j_5,j_6)$, $(j_4,j_2,j_6)$-    and $(j_4,j_5,j_3)$ satisfy the triangle inequalities and have integer sums.-  comment-symmetry: Entries use the lexicographically largest $(j_1,j_2,j_3,j_4,j_5,j_6)$-    among the 24 tetrahedral symmetries of the Wigner $6j$ symbol.-  comment-zeros: Rows whose Wigner $6j$ symbol is zero are omitted. The all-zero symbol-    is also omitted, since its value is $1$.+  comment-symmetry: 'A Wigner $6j$ symbol is unchanged by the 24 tetrahedral symmetries:+    a permutation of the three columns, followed by either no swap or by swaps of+    the upper and lower entries in two columns. Each class is stored once, under the+    lexicographically largest $(j_1,j_2,j_3,j_4,j_5,j_6)$, so a symbol in another+    arrangement is found by applying those symmetries first.'+  comment-admissible: 'A Wigner $6j$ symbol is zero unless each of the four triads+    $(j_1,j_2,j_3)$, $(j_1,j_5,j_6)$, $(j_4,j_2,j_6)$ and $(j_4,j_5,j_3)$ satisfies+    the triangle inequalities and has an integer sum. This condition is necessary+    but not sufficient: of the 490 canonical classes with all $j_i\leq 7/2$, four+    nontrivial admissible classes vanish, including $\begin{Bmatrix}2&2&2\\3/2&3/2&3/2\end{Bmatrix}$.'+  comment-zeros: Rows whose value is zero are omitted, as is the trivial all-zero+    symbol $\begin{Bmatrix}0&0&0\\0&0&0\end{Bmatrix}$, whose value is $1$.   comment-algebraic: Every nonzero Wigner $6j$ symbol is a rational number or a rational-    multiple of the square root of a rational number. Many of the simplest radicals-    are also in HREF{T35}[the table of algebraic numbers of degree 2].+    multiple of the square root of a rational number. Formulas:-  formula-racah: The Racah $W$ coefficient satisfies $\begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\end{Bmatrix}-    =(-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6)$ CITE{Wikipedia}.-  formula-zero: If one entry is zero, then $\begin{Bmatrix}a&b&c\\b&a&0\end{Bmatrix}-    =(-1)^{a+b+c}/\sqrt{(2a+1)(2b+1)}$ CITE{Wikipedia}.-  formula-3j-sum: A Wigner $6j$ symbol is a finite signed sum of products of four-    HREF{T252}[Wigner $3j$ symbols] CITE{Wikipedia}.+  formula-racah: Put $x_1=j_1+j_2+j_3$, $x_2=j_1+j_5+j_6$, $x_3=j_4+j_2+j_6$, $x_4=j_4+j_5+j_3$,+    $y_1=j_1+j_2+j_4+j_5$, $y_2=j_2+j_3+j_5+j_6$ and $y_3=j_3+j_1+j_6+j_4$. With $\Delta(a,b,c)=\sqrt{(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!}$,+    Racah's finite sum is $\begin{aligned}\begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\end{Bmatrix}&=\Delta(j_1,j_2,j_3)\Delta(j_1,j_5,j_6)\Delta(j_4,j_2,j_6)\Delta(j_4,j_5,j_3)\\&\quad\times\sum_s+    \frac{(-1)^s(s+1)!}{\prod_{i=1}^4(s-x_i)!\prod_{r=1}^3(y_r-s)!}.\end{aligned}$+    The sum is over the integers $s$ for which every factorial argument is nonnegative+    CITE{DLMF-6j}.+  formula-zero: If a Wigner $6j$ symbol has an entry $0$, tetrahedral symmetry may+    put that entry in the lower-right position. The value is then zero unless the+    first two columns are a transposed pair, and in the nonzero case $\begin{Bmatrix}j_1&j_2&j_3\\j_2&j_1&0\end{Bmatrix}=(-1)^{j_1+j_2+j_3}/\sqrt{(2j_1+1)(2j_2+1)}$+    CITE{Wikipedia}.+  formula-3j-sum: Equivalently, $\begin{aligned}\begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\end{Bmatrix}&=\sum_{m_1,m_2,m_3,\mu_1,\mu_2,\mu_3}(-1)^{j_4+\mu_1+j_5+\mu_2+j_6+\mu_3}\\&\quad\times\begin{pmatrix}j_1&j_2&j_3\\m_1&m_2&m_3\end{pmatrix}\begin{pmatrix}j_1&j_5&j_6\\m_1&\mu_2&-\mu_3\end{pmatrix}\\&\quad\times\begin{pmatrix}j_4&j_2&j_6\\-\mu_1&m_2&\mu_3\end{pmatrix}\begin{pmatrix}j_4&j_5&j_3\\\mu_1&-\mu_2&m_3\end{pmatrix}.\end{aligned}$+    The sum is over magnetic quantum numbers admissible for all four HREF{T252}[Wigner+    $3j$ symbols] CITE{DLMF-6j}. Programs:   program-sage:

   rigour: proven   complete: 'no'-  complete-note: it holds every nonzero canonical tetrahedral class with all $j_i\leq-    7/2$, except the all-zero symbol-  rigour details: The generator evaluates Racah's finite sum with exact rational arithmetic,-    using integer-scaled angular momenta so every factorial argument is checked before-    it is used. The exact result is a sign times the square root of a rational number,-    converted to an arb ball for the stored real value; rational results are written-    exactly. The values are checked against Richard Mathar's exact `6jSymb` table,-    against the zero-entry formula CITE{formula-zero}, and against the finite sum-    of Wigner $3j$ symbols in CITE{formula-3j-sum}.+  complete-note: it holds every canonical tetrahedral class with all $j_i\leq 7/2$+    whose value is not zero, other than the symbol with every $j_i=0$; this cutoff+    covers the small angular momenta through $7/2$ and gives 485 rows+  rigour details: 'The generator evaluates Racah''s finite sum CITE{formula-racah}+    with exact rational arithmetic, using integer-scaled angular momenta so every+    factorial argument is checked before it is used. The exact result is a sign times+    the square root of a rational number, converted to an arb ball for the stored+    real value; rational results are written exactly.+++    Before any entry is returned, the generator recomputes every row at all 24 tetrahedral+    images and requires the results to agree. It checks the zero-entry formula CITE{formula-zero}+    wherever that formula applies, and it checks every row against the finite sum+    of Wigner $3j$ symbols in CITE{formula-3j-sum}. All 485 stored rows were also+    compared with SageMath''s exact `wigner_6j` function CITE{Sage}.' Display properties:   number-header: $\begin{Bmatrix}j_1&j_2&j_3\\j_4&j_5&j_6\end{Bmatrix}$