Alcove walks: from affine crystals to quantum K-theory

I will present two applications of the alcove model in the
representation
theory of Lie algebras, defined by Gaussent-Littelmann and myself in joint
work with A. Postnikov. The first application is to an efficient
computation, in classical Lie types, of the energy function, which defines
the affine grading on a tensor product of Kirillov-Reshetikhin crystals
(the latter encode certain finite-dimensional representations of quantum
affine algebras as the quantum parameter goes to zero). This application
is based on the Ram-Yip formula for Macdonald polynomials, which is in
terms of the alcove model. The second application is to a conjectured
Chevalley-type multiplication formula in the quantum K-theory of the flag
manifold. A crucial ingredient in both applications is the quantum Bruhat
graph, which is obtained by adding extra edges to the Hasse diagram of the
Bruhat order on a Weyl group. The talk contains joint work with A.
Schilling and T. Maeno.