| We present a number of results concerning crystal bases and their combinatorial models. There are two parts, which are largely independent.;The first part consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and Drinfeld's unitarized R-matrix. We then give a new definition for the crystal commutor, which makes sense for any symmetrizable Kac-Moody algebra. We show that this new definition agrees with A. Henriques and J. Kamnitzer's definition in the finite type case, but we cannot prove our commutor remains a coboundary structure in the non-finite type cases. Next, we give a new formula for the R-matrix and hence for the unitarized R-matrix, which we hope will be useful in eventually proving that the crystal commutor is a coboundary structure in all cases.;In the second part, we define three combinatorial models for sl&d14;n crystals. These are parameterized by partitions, configurations of beads on an "abacus", and cylindric plane partitions, respectively. Our models are reducible, but we can identify an irreducible sub-crystal corresponding to any dominant integral highest weight A. Cylindric plane partitions actually parameterize a basis for an irreducible representation of gl&d14;n . This allows us to calculate the partition function for a system of random cylindric plane partitions first studied by A. Borodin. There is a symmetry in our model which allows us calculate the same partition function using the Weyl character formula of either a level ℓ representation for gl&d14;n or a level n representation for gl&d14;ℓ . Thus we observe a form of rank level duality, originally due to I. Frenkel. Finally, we use an explicit bijection to relate our work to the Kyoto path model. |
