Combinatorics of alcoves with applications to representation theory and KtheoryAlex PostnikovMIT
October 8,

ABSTRACT


Semistandard Young tableaux count characters of irreducible representations of SL_n. Littelmann paths count characters of irreducible representations of an arbitrary semisimple Lie group. However, Littelmann paths are hard to work with. They have much more complicated characterization than, say, Young tableaux. In this talk we present a general simple combinatorial formula for characters of irreducible representations. We also give a Chevalleytype formula for equivariant Ktheory of generalized flag manifolds. Our combinatorial counterpart of a Littelmann path is an alcove path, which a sequence of adjacent alcoves for the affine Weyl group. The construction is given in terms of saturated chains in the Bruhat order. The YangBaxter equation also plays an important role in the construction. This construction is just a tip on an iceberg. Alcoves for the affine Weyl group seem to have very interesting and rich combinatorial structure that is yet to be explored. The talk in based on a joint work with Cristian Lenart. The preprint can be found at arXiv:math.RT/0309207. We will also mention some other results related to combinatorics of alcoves, including joint results with Thomas Lam. The talk should be accessible for graduate students. 
Combinatorics Seminar, Mathematics Department, MIT, sara@math.mit.edu 

