The Symmetry Relations of the12jSymbol

Abstract

A triangular-condition diagram in the form of a Möbius strip is constructed for the $12j$ symbol, and those geometrical transformations of the strip which preserve the conditions are considered. This suggests a revised notation for the $12j$ symbol which exhibits clearly the triangular conditions. The transformations of the strip suggest a complete set of sixteen symmetry relations, which can be verified by use of an explicit expression for the symbol. In the new notation the symmetry relations are obtained easily as simple manipulative rules. The positions of the elements in the symbol also display the difference between the two sets of elements of the symbol with respect to degeneration by setting one element zero.