Three combinatorial models for affine sl(n) crystals, with applications to cylindric plane partitions

Details Three combinatorial models for affine sl(n) crystals, with applications to cylindric plane partitions Three combinatorial models for affine sl(n) crystals, with applications to cylindric plane partitions

2007-02-05

arxiv.org/abs/math/0702062v3

Math. Res. Not. IMRN 2008(2):Art. ID rnm143, 40, 2008

Mathematics

Quantum Algebra

29 pages, 14 Figures. v2: 5 new references. Minor corrections and clarifications. v3: Section 4.2 corrected

Scientific paper

10.1093/imrn/rnm143

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.

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