Three Combinatorial Models for $$ _ $$ Crystals, with Applications to Cylindric Plane Partitions

Abstract

We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parametrize a basis for the tensor product of an irreducible representation with the space spanned by all partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.