An atomic Coxeter presentation
Abstract
We study parabolic double cosets in a Coxeter system by decomposing them into atom(ic coset)s, a generalization of simple reflections introduced in a joint work with Elias, Libedinsky, Patimo. We define and classify braid relations between compositions of atoms and prove a Matsumoto theorem. Together with a quadratic relation, our braid relations give a presentation of nilCoxeter algebroids similar to Demazure's presentation of nilCoxeter algebras. Our consideration of reduced compositions of atoms gives rise to a new combinatorial structure, which is equipped with a length function and a Bruhat order and is realized as Tits cone intersections in the sense of Iyama-Wemyss.
Cite
@article{arxiv.2312.16666,
title = {An atomic Coxeter presentation},
author = {Hankyung Ko},
journal= {arXiv preprint arXiv:2312.16666},
year = {2025}
}
Comments
v3 revision after referee comments which includes some further revision after publication (less relations are needed in Prop. 6.6 and Cor. 6.8). v2 with more extensive intro, updated citation, and minor revisions. 31 pages, 8 color figures