Racah coefficients, subrepresentation semirings, and composite materials

AbstractTypically, 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

The geometry of blueprints. Part I: Algebraic background and scheme theory

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and "archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.Comment: Slightly revised and extended version as in print. 51 page