Character formulae of [formula omitted]n-modules and inhomogeneous paths

Abstract

Let B ( l) be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U q ′( sl ̂ n) . For a partition μ = ( μ 1, …, μ m , elements of the tensor product B ( μ 1) ⊗…⊗ B ( μ m ) can be regarded as inhomogeneous paths. We establish a bijection between a certain large μ limit of this crystal and the crystal of an (generally reducible) integrable U q ′( sl ̂ n)- module , which forms a large family depending on the inhomogeneity of μ kept in the limit. For the associated one-dimensional sums, relations with the Kostka-Foulkes polynomials are clarified, and new fermionic formulae are presented. By combining their limits with the bijection, we prove or conjecture several formulae for the string functions, branching functions, coset branching functions and spinon character formula of both vertex and RSOS types.