Specialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals

The (symmetric) Macdonald polynomials are Weyl group invariant polynomials with rational function coefficients (in q, t), which specialize to the irreducible Lie algebra characters upon setting q = t = 0. Quantum K-theory is a K-theoretic generalization of quantum cohomology. Kirillov-Reshetikhin (KR) modules are certain finite-dimensional modules for affine Lie algebras. 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, we proved that the same specialization of Macdonald polynomials equals the graded character of a tensor product of (one-column) KR modules. I will discuss the combinatorics underlying these connections.