An atomic Coxeter presentation
Hankyung Ko
TL;DR
The paper develops an atomic combinatorics for parabolic double cosets in Coxeter systems by factoring core cosets into atoms and establishing atomic braid relations. It proves an atomic Matsumoto theorem, giving a robust, width-preserving equivalence among atomic reduced expressions, and provides a presentation of core cosets via an atomic (nilCoxeter) Demazure-like framework. This atomic approach yields a natural nilCoxeter algebroid presentation and clarifies how core cosets sit inside singular Coxeter structures, extending to nonregular types. Moreover, the authors connect the atomic theory to Lie-theoretic contexts via the singular Hecke category and to geometric/combinatorial frameworks of Tits cone intersections as studied by Iyama–Wemyss, enriching both combinatorics and representation-theoretic applications.
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.