Subword complexes and 2-truncated cubes

TL;DR

This paper explores a family of subword complexes related to Coxeter groups, showing their dual polytopes are combinatorial 2-truncated cubes, thus generalizing known cluster complexes.

Contribution

It introduces a new family of subword complexes parameterized by reduced expressions, linking them to root systems and proving their duals are 2-truncated cubes.

Findings

Vertices described via roots of the root system

Dual polytopes are combinatorial 2-truncated cubes

Generalizes c-cluster complexes

Abstract

For a Coxeter element $c$ of a finite Coxeter group, we consider a family of subword complexes parameterized by reduced expressions of the longest element. This family generalizes $c-$cluster complexes. We describe vertices of these complexes in terms of roots of the corresponding root system. We prove that dual polytopes of all such complexes are combinatorial 2-truncated cubes.