Angular momentum theory, umbral calculus, and combinatorics

The theory of the unitary irreducible representations of the unitary group SU(2) is reviewed with the objective of demonstrating the fundamental role that the umbral calculus and combinatorics play. For the physicist, the study of the group SU(2) is synonymous with the theory of angular momentum in quantum theory, and Kronecker products of irreducible unitary representations comprise the mathematical apparatus for building composite systems from simpler constituents. The Clebsch-Gordon coefficients, which are essential to the binary theory of composite systems, are shown to have an umbral calculus origin. MacMahonʹs master theorem is shown to be the basic result for generating the mathematical quantities needed for bringing comprehension to the properties of composite systems.