Counting chains in the noncrossing partition lattice via the W-Laplacian

Abstract

We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^{n}n!/|W|$ for the number of maximal chains in the noncrossing partition lattice $NC(W)$ of a real reflection group $W$. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the $W$-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

arXiv comment17 pages, comments very much welcome!

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Cited by in corpus (6)

• Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups
• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• The diagonal coinvariant ring of a complex reflection group
• Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
• Bijection between positive clusters and projectively signed exceptional sequences
• Hurwitz numbers for reflection groups III: Uniform formulas