A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$

Abstract

We establish the equality of the specialization |$P_\lambda(x;q,0)$| of the Macdonald polynomial at |$t=0$| with the graded character |$X_\lambda(x;q)$| of a tensor product of “single-column” Kirillov–Reshetikhin modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated with Macdonald polynomials) and the quantum Lakshmibai–Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the |$t = 0$| limit of the symmetric Macdonald polynomial in the untwisted and non-simply laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.