Kirillov-Reshetikhin crystals, Macdonald polynomials, affine Demazure characters, and combinatorial models

In recent work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we show that in all untwisted affine types the specialization of a Macdonald polynomial at t=0 is the graded character of a tensor product of one-column Kirillov-Reshetikhin (KR) modules. We also obtain two uniform models for the corresponding KR crystals, namely a generalization of the Lakshmibai-Seshadri (canonical Littelmann) paths based on the so-called parabolic quantum Bruhat graph, and the quantum alcove model of myself and A. Lubovsky. I will also mention other closely related topics: affine Demazure crystals (extending the work of Ion and Fourier-Littelmann), expressing the energy function, and a uniform realization of the combinatorial R-matrix, which commutes factors in a tensor product of KR crystals (with A. Lubovsky).