Crystal Duality and Littlewood-richardson Rule of Extremal Weight Crystals

We consider a category of gl∞-crystals, whose object is a disjoint union of extremal weight crystals with bounded non-negative level and finite multiplicity for each connected component. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is antiisomorphic to an Ore extension of the character ring of integrable lowest gl∞modules with respect to derivations shifting the fundamental weight characters. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients. A double crystal structure on binary matrices of various shapes and associated crystal dualities are used as our main tools.