9-j Symbol - Symmetry Relations

Symmetry Relations

A symbol is invariant under reflection in either diagonal:

\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_4 & j_7\\ j_2 & j_5 & j_8\\ j_3 & j_6 & j_9 \end{Bmatrix} = \begin{Bmatrix} j_9 & j_6 & j_3\\ j_8 & j_5 & j_2\\ j_7 & j_4 & j_1 \end{Bmatrix}.

These equations represent two symmetry operations of the associated Yutsis graph on 6 nodes.

The permutation of any two rows or any two columns yields a phase factor, where

S=\sum_{i=1}^9 j_i.

For example:

\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_4 & j_5 & j_6\\ j_1 & j_2 & j_3\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6\\ j_8 & j_7 & j_9 \end{Bmatrix}.

There are 6 possible permutations of three rows, 6 possible permutations of three columns, and together with the 2 symmetries related to the diagonals these account for the 72 = 2*6*6 symmetry operations of the associated automorphism group of the graph.

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