Séminaire Lotharingien de Combinatoire, 93B.73 (2025), 12 pp.
Clément Chenevière, Wenjie Fang and Corentin Henriet
A New Definition for m-Cambrian Lattices
Abstract.
The Cambrian lattices, introduced in (Reading, 2006), generalize the Tamari lattice to any choice of Coxeter element in any finite Coxeter group.
They are further generalized to the m-Cambrian lattices (Stump, Thomas, Williams, 2015).
However, their definitions do not provide a practical setup to work with combinatorially.
In this paper, we provide a new equivalent definition of the m-Cambrian lattices on simple objects called m-noncrossing partitions, using a simple and effective comparison criterion.
It is obtained by showing that each interval has a unique maximal chain that is \textsf{c}-increasing, which is computed by a greedy algorithm.
Our proof is uniform, involving all Coxeter groups and all choices of Coxeter element at the same time.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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