On Quasi-Ambivalent Groups

Abstract

The prototype for applications of group theory to physics, and to mathematical physics, is the quantum theory of angular momentum [1] ; the use of such techniques is now almost universal, and familiarly (through somewhat imprecisely) known as “Racah algebra”. To categorize, group theoretically, those characteristics which underlay this applicability to physical problems, Wigner [30] isolated two significant conditions, and designated groups possessing these properties as simply reducible.The two conditions for simple reducibility are:(a)Every element is equivalent to its reciprocal, i.e., all classes are ambivalent.(b) The Kronecker (or “direct“) product of any two irreducible representations of the group contains no representation more than once.