Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups

TL;DR

This paper explores the geometric and combinatorial structures of Bruhat intervals and subword complexes in finite Coxeter groups, introducing brick polyhedra that unify these concepts.

Contribution

It establishes new connections between Bruhat intervals, subword complexes, and root configurations, leading to the definition of brick polyhedra for subword complexes.

Findings

Connected cones of Bruhat intervals and subword complexes.

Introduced brick polyhedra as a unifying geometric object.

Enhanced understanding of Coxeter group combinatorics.

Abstract

We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.