Structure of the Crossing Matrix for Arbitrary Internal Symmetry Groups

Some properties of crossing matrices are deduced which are independent of the particular symmetry group from which the crossing matrices are derived. In particular, a, factorization of any elastic crossing matrix, analogous to the factorization of a rotation matrix in n dimensions is found, and the connection between, crossing and unitary matrices elucidated. In the special case of the crossing matrix for elastic scattering of particles which transform as representations of ${\mathrm{SU}}_2$ , as well as the usual consistency requirement that each row should sum to unity, a new consistency requirement on the elements in a given column is proved. As a byproduct of this work, a possibly new quadratic identity for Racah coefficients is exhibited.