Racah coefficients, subrepresentation semirings, and composite materials

Typically, physical properties of composite materials are strongly dependent on microstructure. However, in exceptional situations, exact relations exist which are microstructure-independent. Grabovsky has constructed an abstract theory of exact relations, reducing the search for exact relations to a purely algebraic problem involving the multiplication of SO (3)-subrepresentations in certain endomorphism algebras. This motivates us to introduce subrepresentation semirings, algebraic structures which formalize subrepresentation multiplication. 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.