Racah coefficients, subrepresentation semirings, and composite materials

We study the ideals and subsemirings of these semirings, relating them to properties of the underlying G-algebra and proving classification theorems in the case of endomorphism algebras of representations. For SU(2), we compute these semirings for general V. When V is irreducible, we describe the semiring structure explicitly in terms of the vanishing of Racah coefficients, coefficients familiar from the quantum theory of angular momentum. In fact, we show that Racah coefficients can be defined entirely in terms of subrepresentation multiplication.