Continuous Noncrossing Partitions and Weighted Circular Factorizations

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.