A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph

TL;DR

This paper introduces a unified framework connecting the parabolic quantum Bruhat graph with affine Weyl groups and Littelmann's poset, enabling a consistent construction of KR crystals and relating Macdonald polynomials to KR modules.

Contribution

It lifts the parabolic quantum Bruhat graph into affine structures and generalizes key lemmas, facilitating a uniform approach to KR crystals and Macdonald polynomial relations.

Findings

Established a quantum analogue of Deodhar's Bruhat-minimum lift.

Generalized Postnikov's lemma to the parabolic quantum Bruhat graph.

Set the stage for a uniform construction of tensor products of KR crystals.

Abstract

We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph 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.