The DARK side of generalized Demazure crystals

TL;DR

This paper extends Naoi's work on tensor products of perfect Kirillov-Reshetikhin crystals, exploring distinguished subsets and their combinatorial descriptions, leading to progress on longstanding conjectures in type A.

Contribution

It generalizes Naoi's results to specific subsets of tensor products and connects them to katabolizable tableaux, advancing understanding of Demazure crystals in type A.

Findings

Extended Naoi's isomorphism results to distinguished subsets

Connected subsets to katabolizable tableaux in type A

Resolved conjectures of Shimozono-Weyman and Chen-Haiman

Abstract

Naoi showed that tensor products of perfect Kirillov-Reshetikhin crystals are isomorphic to certain generalized Demazure crystals. We extend Naoi's results to address distinguished subsets of these tensor products. In type A, these are naturally described in terms of katabolizable tableaux which was key to resolving conjectures of Shimozono-Weyman and Chen-Haiman in arXiv:2007.04952.