Crystal Structure of Localized Quantum Unipotent Coordinate Category
Abstract
A localized quantum unipotent coordinate category associated with a Weyl group element is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in possesses a crystal structure and it is isomorphic to the cellular crystal associated with . As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal.
Keywords
Cite
@article{arxiv.2502.14319,
title = {Crystal Structure of Localized Quantum Unipotent Coordinate Category},
author = {Masaki Kashiwara and Toshiki Nakashima},
journal= {arXiv preprint arXiv:2502.14319},
year = {2025}
}
Comments
In v1, Proposition 6.8 is false. There is a counterexample: $\mathfrak{g}=A_3$, $i=2$, $w=s_1s_3s_2s_1s_3$. Since the paper heavily depends on this proposition, we withdraw v1. We do not know if Theorem 6.12 is true or not. In v3 with 62 pages, the error is corrected