Spin Module Versions of Weyl's Reciprocity Theorem for Classical Kac-Moody Lie Algebras — An Application to Branching Rule Duality

Abstract

We study tensor products of the spin modules (i.e. the Fermion Fock space representations) for classical (simple or affine) Kac-Moody Lie algebras. We find out that there are mutually commutant pairs of classical Kac-Moody algebras acting on the spin modules, and describe the irreducible decompositions in terms of Young diagrams. As applications, we obtain a simple explanation of Jimbo-Miwa's branching rule duality (i.e. isomorphisms between coset Virasoro modules) [JM], generalization thereof and the duality of the modular transformation rules of affine Lie algebra characters