The Racah–Wigner category

Abstract

The Racah–Wigner calculus is formulated using category theory. The notion of recoupling requires a consistent choice of isomorphisms corresponding to regrouping of irreducible representations. This is the essential content of a ring category. Hence the Racah–Wigner calculus is inherently a ring category. Category theory places the emphasis on maps between representations. The diagrammatic approach of category theory lays bare underlying relationships in the Racah–Wigner calculus. Direct sum decomposition, coupling coefficients, recoupling coefficients, and group chain decomposition are simplified and clarified when represented by diagrams. The diagram techniques of category theory are unlike diagram techniques used for group calculations. The Biedenharn–Elliott sum rule, Racah backcoupling, Racah factorisation lemma, and the Wigner–Eckart theorem are derived from properties of this category. PACS Nos.: 02.10Ws, 02.20Qs, 02.20Fh, 02.20Df, 03.65Fd, 31.15Hz