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arXiv · 2109.04341

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

Guillaume Chapuy·Theo Douvropoulos
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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.

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Guillaume Chapuy, Theo Douvropoulos. 2021-09-09. Counting chains in the noncrossing partition lattice via the W-Laplacian. https://doi.org/10.1016/j.jalgebra.2022.02.023

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