Journal
JOURNAL OF ALGEBRA
Volume 602, Issue -, Pages 381-404Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jalgebra.2022.02.023
Keywords
Noncrossing partitions; Coxeter elements; W-Laplacian; Artin groups; Coxeter-Catalan combinatorics; Dual braid presentation
Categories
Funding
- European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme [ERC-2016-STG 716083 CombiTop]
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This article presents an elementary derivation based on Coxeter theory to calculate the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. By comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial, we provide a proof and discuss the implications of this formula for the geometric group theory of spherical and affine Artin groups.
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 DeligneReading 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.(C) 2022 Elsevier Inc. All rights reserved.
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