Symmetry Properties of the Wigner9jSymbol

Abstract

The 72 symmetry relations of the Wigner $9j$ symbol $\left\{{{j}_{1},{j}_{2},{j}_{3}}{{j}_{4},{j}_{5},{j}_{6}}{{j}_{7},{j}_{8},{j}_{9}}\right\}=\frac{〈({j}_{1}{j}_{2}){j}_{3}, ({j}_{4}{j}_{5}){j}_{6}, {j}_{9}m|({j}_{1}{j}_{4}){j}_{7}, ({j}_{2}{j}_{5}){j}_{8}, {j}_{9}m〉}{{{(2{j}_{3}+1)(2{j}_{6}+1)(2{j}_{7}+1)(2{j}_{8}+1)}}^{\frac{1}{2}}}$ are given. The group of symmetry may be generated by the following elements: (i) an odd permutation of the rows or columns of the symbol multiples it by ${(\ensuremath{-}1)}^{R}$, where $R={j}_{1}+{j}_{2}+{j}_{3}+{j}_{4}+{j}_{5}+{j}_{6}+{j}_{7}+{j}_{8}+{j}_{9}$; (ii) a reflection of the symbol in either of the two diagonals leaves it invariant. These results are conveniently deduced from the Schwinger generating function for the $9j$ symbol. In an addendum a $12j$ symbol is defined and some of its properties discussed.