6-j Symbol - Symmetry Relations

Symmetry Relations

The 6-j symbol is invariant under the permutation of any two columns:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_3 & j_2\\ j_4 & j_6 & j_5 \end{Bmatrix} = \begin{Bmatrix} j_3 & j_2 & j_1\\ j_6 & j_5 & j_4 \end{Bmatrix}.$

The 6-j symbol is also invariant if upper and lower arguments are interchanged in any two columns:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_5 & j_3\\ j_1 & j_2 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_5 & j_6\\ j_4 & j_2 & j_3 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_2 & j_6\\ j_1 & j_5 & j_3 \end{Bmatrix}.$

These equations reflect the 24 symmetry operations of the automorphism group that leave the associated tetrahedral Yutsis graph with 6 edges invariant: mirror operations that exchange two vertices and a swap an adjacent pair of edges.

The 6-j symbol

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix}$

is zero unless j₁, j₂, and j₃ satisfy triangle conditions, i.e.,

$j_1 = |j_2-j_3|, \ldots, j_2+j_3.$

In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for (j₁, j₅, j₆), (j₄, j₂, j₆), and (j₄, j₅, j₃).

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