Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups
Abstract
The interval [1,c]T between the identity element and a Coxeter element c in the absolute order on a Coxeter group W is a generalization of the poset of noncrossing partitions arising when W is the symmetric group. When W is finite, this poset is always a lattice, and it is natural to associate to every element w∈ [1,c]T its \textit{moved space} Mov(w)=Im(w − IdV) in the geometric representation V of W. It has dimension equal to the reflection length ℓT(w) of w, and gives a realization of [1,c]T inside the lattice of subspaces of V. It is an important tool in the study of [1,c]T. When W is infinite, the moved space of an element w∈ [1,c]T no longer has dimension ℓT(w) in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element w∈ [1,c]T in an arbitrary Coxeter group, that we call \textit{absolute moved space} of w. This subspace AM(w) of V always contains Mov(w) and has dimension equal to ℓT(w), and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from [1,c]T to reflection subgroups of W, which to w∈ [1,c]T associates the subgroup P(w) generated by reflections lying below w in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of [1,c]T when W has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which [1,c]T fails to be a lattice.
Related
- Noncrossing combinatorics of quasi-Coxeter elements
- Grassmannians of absolute Banach-Colmez Spaces
- Symmetric products and finite subset spaces of absolute Lipschitz retracts
- Partition invariance of variation indices and classical p-variation spaces
- The $2$-co-lacunary subsequence principle for arbitrary noncommutative L1-spaces
BibTeX
@misc{gobet2026absolute,
title = {{Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups}},
author = {Thomas Gobet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37867},
eprint = {2609.37867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37867},
}
RIS
TY - UNPB
TI - Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups
AU - Thomas Gobet
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37867
UR - https://arxiv.org/abs/2609.37867
AB - The interval [1,c]_T between the identity element and a Coxeter element c in the absolute order on a Coxeter group W is a generalization of the poset of noncrossing partitions arising when W is the symmetric group. When W is finite, this poset is always a lattice, and it is natural to associate to every element w∈ [1,c]_T its \textit{moved space} Mov(w)=Im(w − Id_V) in the geometric representation V of W. It has dimension equal to the reflection length ℓ_T(w) of w, and gives a realization of [1,c]_T inside the lattice of subspaces of V. It is an important tool in the study of [1,c]_T. When W is infinite, the moved space of an element w∈ [1,c]_T no longer has dimension ℓ_T(w) in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element w∈ [1,c]_T in an arbitrary Coxeter group, that we call \textit{absolute moved space} of w. This subspace AM(w) of V always contains Mov(w) and has dimension equal to ℓ_T(w), and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from [1,c]_T to reflection subgroups of W, which to w∈ [1,c]_T associates the subgroup P(w) generated by reflections lying below w in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of [1,c]_T when W has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which [1,c]_T fails to be a lattice.
ER -