Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Abstract

We consider a category of gl ∞ -crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight gl ∞ -modules with respect to derivations shifting the characters of fundamental weight modules. A Littlewood–Richardson rule of extremal weight crystals of non-negative level is described explicitly in terms of classical Littlewood–Richardson coefficients.