Combinatorics of alcoves with applications to representation theory and K-theory

Semistandard Young tableaux count characters of irreducible representations of SL_n. Littelmann paths count characters of irreducible representations of an arbitrary semi-simple 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 Chevalley-type formula for equivariant K-theory 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 Yang-Baxter 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.