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

Abstract

We consider two lifts of the parabolic quantum Bruhat graph, one into the Bruhat order in the affine Weyl group and the other into a level-zero weight poset first considered by Littelmann. The lift into the affine Weyl group gives rise to Diamond Lemmas for the parabolic quantum Bruhat graph. Littlemann's poset is defined on Lakshmibai-Seshadri paths for arbitrary (not necessarily dominant) weights. Here we consider this poset for level-zero weights and determine its local structure (such as cover relations) in terms of the parabolic quantum Bruhat graph. Littelmann had not determined the local structure of this poset. In addition, we show a generalization of results by Deodhar, coined the tilted-Bruhat theorem, which involves the compatibility of the quantum Bruhat graph with the cosets for every parabolic subgroup of the Weyl group. We will use the results in this paper in a second paper to establish the equality between the Macdonald polynomials $P_\lambda (q,t)$ specialized at $t = 0$ and $X_\lambda (q)$ which is the graded character of a simple Lie algebra coming from tensor products of Kirillov-Reshetikhin (KR) modules.