6-j symbol

When j6 = 0 the expression for the 6-j symbol is: The triangular delta {j1  j2  j3} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and zero otherwise.

[3] The asymptotic formula applies when all six quantum numbers j1, ..., j6 are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron.

In representation theory, 6-j symbols are matrix coefficients of the associator isomorphism in a tensor category.

Then: The associativity isomorphism induces a vector space isomorphism and the 6j symbols are defined as the component maps: When the multiplicity spaces have canonical basis elements and dimension at most one (as in the case of SU(2) in the traditional setting), these component maps can be interpreted as numbers, and the 6-j symbols become ordinary matrix coefficients.

For the case of representations of a finite group, it is well known that the character table alone (which determines the underlying abelian category and the Grothendieck ring structure) does not determine a group up to isomorphism, while the symmetric monoidal category structure does, by Tannaka-Krein duality.

Jucys diagram for the Wigner 6- j symbol. The plus sign on the nodes indicates an anticlockwise reading of its surrounding lines. Due to its symmetries, there are many ways in which the diagram can be drawn. An equivalent configuration can be created by taking its mirror image and thus changing the pluses to minuses.