Continuous Noncrossing Partitions and Weighted Circular Factorizations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918076734242816 |
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| author | Dougherty, Michael McCammond, Jon |
| author_facet | Dougherty, Michael McCammond, Jon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous Noncrossing Partitions and Weighted Circular Factorizations Dougherty, Michael McCammond, Jon Group Theory Combinatorics Geometric Topology 20F36, 20F65, 05E45, 30C10 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. |
| title | Continuous Noncrossing Partitions and Weighted Circular Factorizations |
| topic | Group Theory Combinatorics Geometric Topology 20F36, 20F65, 05E45, 30C10 |
| url | https://arxiv.org/abs/2507.00283 |