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### Crystal bases and Geometric crystals

**Algebra & Discrete Mathematics**

Speaker: | Arkady Berenstein, University of Oregon |

Location: | 693 Kerr |

Start time: | Fri, Oct 24 2003, 2:10PM |

Crystal bases were introduced by M. Kashiwara as a combinatorial model for good bases in representations of Lie groups and their quantum counterparts.

My talk is based on the results of joint work with David Kazhdan, in which work we proposed a new construction of certain Kashiwara's crystal bases, including all the finite irreducible ones.

Geometric crystals (which play the main part in the construction) allow us to study the crystal bases both geometrically (i.e., in terms of rational morphisms of algebraic varieties) and combinatorially (i.e., in terms of piecewise-linear maps of polyhedral sets). Quite unexpectedly, the passage from geometric crystals to crystal bases requires some kind of the Langlands duality.

The purely geometric approach to crystal bases also reveals some hidden combinatorial structures: if we denote by $B_0$ the union of all finite irreducible crystal bases, then our construction gives a ''crystal multiplication'' on $B_0$ and a ``central charge" function on the square of $B_0$.