Racah Coefficients, Subrepresentation Semirings, and Composite Materials

Typically, physical properties of composite materials are strongly dependent on the microstructure. However, in exceptional situations, there exists exact formulas for the effective properties of composites which are microstructure-independent. These formulas represent fundamental physical invariances. The classical approach to exact relations suffered from the drawback that the methods used were heavily dependent on the physical context. In the late 1990’s, Grabovsky constructed an abstract theory of exact relations. This theory reduced the search for exact relations to a purely algebraic problem involving the multiplication of subrepresentations of the rotation group SO(3) in the space of linear operators on certain tensor spaces. This motivates us to introduce subrepresentation semirings, algebraic structures which formalize the multiplication of subrepresentations. We show that there is a one-to-one correspondence between the invariant ideals and subalgebras of a G-algebra and the saturated ideals and subsemirings of the associated subrepresentation semiring. When the G-algebra is the endomorphism algebra of a representation, we classify the saturated ideals in general and the saturated subsemirings under the assumptions that the ground field is algebraically closed and the underlying representation is irreducible. For SU(2), we compute these semirings for an arbitrary complex finite-dimensional representation; this includes the cases of interest for applications to composite materials. When the representation is irreducible, we show that the subrepresentation semiring can be described explicitly in terms of the vanishing of Racah coefficients, coefficients which are familiar from the quantum theory of angular momentum. We also show that Racah coefficients can be defined entirely in terms of multiplication of subrepresentations. 1. Exact relations–A problem from the theory of