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Some Results on the Crystal Commutor and SL(N) Crystals

Some Results on the Crystal Commutor and SL(N) Crystals


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About the Book

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.


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Product Details
  • ISBN-13: 9781243540249
  • Publisher: Proquest, Umi Dissertation Publishing
  • Publisher Imprint: Proquest, Umi Dissertation Publishing
  • Height: 246 mm
  • Weight: 259 gr
  • ISBN-10: 1243540249
  • Publisher Date: 01 Sep 2011
  • Binding: Paperback
  • Spine Width: 8 mm
  • Width: 189 mm


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