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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