Journal of Lie Theory
Vol. 15, No. 1, pp. 321–333 (2005)
Quantum Racah Coefficients and Subrepresentation Semirings
Daniel S. SageDaniel S. Sage
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA
Abstract: Let $G$ be a group and $A$ a $G$-algebra. The subrepresentation semiring of $A$ is the set of subrepresentations of $A$ endowed with operations induced by the algebra operations. The introduction of these semirings was motivated by a problem in material science. 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 product of SU$(2)$-subrepresentations in certain endomorphism algebras. We have shown that the structure of the associated semirings can be described explicitly in terms of Racah coefficients. In this paper, we prove an analogous relationship between Racah coefficients for the quantum algebra U and semirings for endomorphism algebras of representations of U. We generalize the construction of subrepresentation semirings to the Hopf algebra setting. For U, we compute these semirings for the endomorphism algebra of an arbitrary complex finite-dimensional representation. When the representation is irreducible, we show that the subrepresentation semiring can be described explicitly in terms of the vanishing of $q$-Racah coefficients. We further show that $q$-Racah coefficients can be defined entirely in terms of the multiplication of subrepresentations.
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