Journal of Combinatorics

Volume 9 (2018)

Number 2

Chains in shard intersection lattices and parabolic support posets

Pages: 309 – 325

DOI: https://dx.doi.org/10.4310/JOC.2018.v9.n2.a5

Authors

Pierre Baumann (Institut de Recherche Mathématique Avancée, Université de Strasbourg, France)

Frédéric Chapoton (Institut de Recherche Mathématique Avancée, Université de Strasbourg, France)

Christophe Hohlweg (LaCIM et Département de Mathématiques, Université du Québec à Montréal, Canada)

Hugh Thomas (LaCIM et Département de Mathématiques, Université du Québec à Montréal, Canada)

Abstract

For every finite Coxeter group $W$, we prove that the number of chains in the shard intersection lattice introduced by Reading and in the parabolic support poset introduced by Bergeron, Zabrocki and the third author, are the same. We also show that these two partial orders are related by an equality between generating series for their Möbius numbers, and provide a dimension-preserving bijection between the order complex on the parabolic support poset and the pulling triangulation of the permutahedron arising from the right weak order, analogous to the bijection defined by Reading between the order complex of the shard intersection order and the same triangulation of the permutahedron.

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P. Baumann a été soutenu par l’Agence Nationale de la Recherche (projets Vargen et GeoLie, références ANR-13-BS01-0001-01 et ANR-15-CE40-0012).

F. Chapoton a été soutenu par l’Agence Nationale de la Recherche (projet Carma, référence ANR-12-BS01-0017).

C. Hohlweg was supported by NSERC Discovery grant Coxeter groups and related structures.

H. Thomas was supported by an NSERC Discovery grant and the Canada Research Chairs program.

Received 24 October 2016

Published 22 January 2018