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

TL;DR

This paper provides a Coxeter-theoretic derivation of the number of maximal chains in the noncrossing partition lattice of a reflection group, linking combinatorics, algebra, and geometry.

Contribution

It introduces an elementary, case-free proof of a key formula for counting chains in noncrossing partitions using the W-Laplacian and compares recursion methods.

Findings

Derived a new proof for the count of maximal chains in NC(W)

Connected the formula to the characteristic polynomial of the W-Laplacian

Discussed implications for spherical and affine Artin groups

Abstract

We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^nn!/|W|$ 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.