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.

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