Affine Lie algebras, quantum K-theory, and combinatorics

Kirillov-Reshetikhin (KR) modules are certain finite-dimensional modules for affine Lie algebras. The (symmetric) Macdonald polynomials are Weyl group invariant polynomials with rational function coefficients in q,t, which specialize to the irreducible characters of simple Lie algebras upon setting q=t=0. Quantum K-theory is a K-theoretic generalization of quantum cohomology. Braverman and Finkelberg related the Macdonald polynomials specialized at t=0 to the quantum K-theory of flag varieties. With S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, I proved that the same specialization of Macdonald polynomials equals the graded character of a tensor product of (one-column) KR modules. I will discuss a combinatorial model which underlies these connections and related computations. The talk is largely self-contained.