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Three combinatorial models for $\widehat{sl}_n$ crystals, with applications to cylindric plane partitions

Algebra & Discrete Mathematics

Speaker: Peter Tingley, UC-Berkeley
Location: 1147 MSB
Start time: Thu, Feb 8 2007, 3:10PM

We define three combinatorial models for $\widehat{sl}_n$ 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 $\Lambda$. Cyclic plane partitions actually parametrize a basis for $V_\Lambda \otimes F$, where $F$ is the space spanned by partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of weight level duality.