A Uniform Model for Kirillov-Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

We lift the parabolic quantum Bruhat graph (QBG) into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analog of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the QBG on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the QBG to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules.