Continuous Noncrossing Partitions and Weighted Circular Factorizations

2025/06/30 by Michael R. Dougherty, Jon McCammond, Dougherty, Michael +1
Mathematics · #05E45 #20F36 #20F65 #30C10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2507.00283

openalex publication_date 2025/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article examines noncrossing partitions of the unit circle in the complex plane; we call these continuous noncrossing partitions. More precisely, we focus on the degree-d continuous noncrossing partitions where unit complex numbers in the same block have identical d-th powers. We prove that the degree-d continuous noncrossing partitions form a topological poset whose uncountable set of elements can be indexed by equivalence classes of objects we call weighted linear factorizations of factors of a d-cycle. Moreover, the maximal elements in this poset form a subspace homeomorphic to the dual Garside classifying space for the d-strand braid group. The degree-d continuous noncrossing partitions of the unit circle are a special case of a more general construction. For every choice of Coxeter element c in any Coxeter group W we define a topological poset of equivalence classes of weighted linear factorizations of factors of c in W whose elements we call continuous c-noncrossing partitions. The maximal elements in this poset form a subspace homeomorphic to the one-vertex complex whose fundamental group is the corresponding dual Artin group.

Citations

• Geometric Combinatorics of Polynomials I: The Case of a Single Polynomial
• The seven-strand braid group is CAT(0)
• Critical Points, Critical Values, and a Determinant Identity for Complex Polynomials
• Degree-d-invariant laminations
• Non-crossing partitions
• Noncrossing hypertrees
• Factoring euclidean isometries
• Dual euclidean Artin groups and the failure of the lattice property
• The structure of euclidean Artin groups
• Artin groups of euclidean type
• The 6-strand braid group is CAT(0)
• Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
• Noncrossing partitions in surprising locations
• The dual braid monoid

Related

• Non-crossing partitions
• Groupes de réflexion, géométrie du discriminant et partitions non-croisées
• Noncrossing partitions of an annulus
• Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
• Charmed roots and the Kroweras complement