Group Representations and Geometry

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Abstract
We confine our attention here to simply reducible groups and show how six of the seven points of a finite projective plane PG(2, 2) constitute a ``Pasch'' diagram representing a 6j symbol. The class of all equivalent symbols may thus be represented by the seventh point in the plane. Analyzing the symmetries of such configurations, we derive two theorems, the first of which is the geometrical analog of Regge's result while the second gives the geometrical analog of the multiplication of two 6j symbols. In these terms the analogs of Eqs. (I1), (I2), and (I3) of Appendix I of Irreducible Tensorial Sets by Fano and Racah are very simply expressed. In particular, the Biedenharn identity (I3) becomes a vector equation (mod 2), and the relation with Desargues' theorem is clarified. The advantage of this geometrical model is that the structure alone survives and all summations and complicated coefficients disappear.

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