Convex realizations of spherical subword complexes
Let $(W, S)$ be a finite Coxeter system with length function $\ell$, let $Q = (q_1, \dots, q_m)$ be a word in $S$ and let $w \in W$. The subword complex $\Delta(Q, w)$ is the simplicial complex whose facets are the sets $I \subseteq \{1, \dots, m\}$ for which the subword of $Q$ at the positions outside $I$ is a reduced expression for $w$. Knutson and Miller proved that subword complexes are vertex decomposable, and that a nonempty one is a ball or a sphere, a sphere exactly when the Demazure product of $Q$ is $w$. Ceballos, Labbé and Stump showed that every spherical subword complex of type $W$ is a link of a face of a multi-cluster complex of type $W$, so that all spherical subword complexes of type $W$ are polytopal if and only if all multi-cluster complexes of type $W$ are; the cluster complexes among them are realized by the generalized associahedra of Chapoton, Fomin and Zelevinsky. Pilaud and Stump realized the spherical subword complexes of root-independent words by the polar duals of brick polytopes, and Bossinger, Gorsky and Simental realized those of double root free words, for Weyl groups of simple Lie groups (2026).
Problem. Suppose that $\Delta(Q, w)$ is homeomorphic to a sphere of dimension $m - \ell(w) - 1 \ge 0$. Must it be combinatorially isomorphic to the boundary complex of a simplicial convex polytope of dimension $m - \ell(w)$?
Source: Knutson and Miller (2004), Question 6.4, for all Coxeter groups. Stated for finite Coxeter systems by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- GR, MG
- Keywords
- AIM Open Applied Problems, Coxeter groups, simplicial polytopes, subword complexes
References
@article{knutson2004,
author = {Knutson, Allen and Miller, Ezra},
title = {Subword complexes in {C}oxeter groups},
journal = {Adv. Math.},
volume = {184},
number = {1},
year = {2004},
pages = {161--176},
doi = {10.1016/S0001-8708(03)00142-7},
eprint = {math/0309259},
archivePrefix = {arXiv},
}
@article{ceballos2014,
author = {Ceballos, Cesar and Labb{\'e}, Jean-Philippe and Stump, Christian},
title = {Subword complexes, cluster complexes, and generalized multi-associahedra},
journal = {J. Algebraic Combin.},
volume = {39},
number = {1},
year = {2014},
pages = {17--51},
doi = {10.1007/s10801-013-0437-x},
eprint = {1108.1776},
archivePrefix = {arXiv},
}
@article{chapoton2002,
author = {Chapoton, Fr{\'e}d{\'e}ric and Fomin, Sergey and Zelevinsky, Andrei},
title = {Polytopal realizations of generalized associahedra},
journal = {Canad. Math. Bull.},
volume = {45},
number = {4},
year = {2002},
pages = {537--566},
doi = {10.4153/CMB-2002-054-1},
eprint = {math/0202004},
archivePrefix = {arXiv},
}
@article{pilaud2015,
author = {Pilaud, Vincent and Stump, Christian},
title = {Brick polytopes of spherical subword complexes and generalized associahedra},
journal = {Adv. Math.},
volume = {276},
year = {2015},
pages = {1--61},
doi = {10.1016/j.aim.2015.02.012},
eprint = {1111.3349},
archivePrefix = {arXiv},
}
@misc{bossinger2026,
author = {Bossinger, Lara and Gorsky, Mikhail and Simental, Jos{\'e}},
title = {Torus actions on compactified braid varieties and polytopality of subword complexes},
year = {2026},
eprint = {2609.12414},
archivePrefix = {arXiv},
}
@misc{aim-subword-polytopality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Convex realizations of spherical subword complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/624-subword-polytopality.md},
note = {Entry 624 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00906, "Convex realizations of spherical subword complexes", clopen.solutions.