A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum bruhat graph
Abstract
© 2014 © The Author(s) 2014. Published by Oxford University Press. All rights reserved. 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.