Racah–Wigner approach to standardization of permutation representations for finite groups

The problem of equivalence of permutation representations of the finite groups is discussed in terms of transforms by bijections of the carrier sets and by group automorphisms. A formal description of a transformation between equivalent representations is given, and a standard form for an arbitrary permutation representation is proposed. The standardization is achieved through the canonical realization of transitive representations and of imprimitivity sets on the left cosets of the group with respect to an appropriate stability subgroup. The purpose of this paper is to pave the way for a systematic formulation of permutation representations, analogous to the Racah algebra of angular momentum theory, which will be useful to multicentre problems of quantum mechanics and statistical physics.