A new definition for m-Cambrian lattices

TL;DR

The authors provide a practical, uniform definition of the $m$-Cambrian lattice using $m$-noncrossing partitions, via a simple vertical/diagonal comparability rule and a greedy ${\sf c}$-increasing chain computation that works across all finite Coxeter groups and Coxeter elements. They establish equivalence with the existing ${\sf Sort}$, ${\sf SC}$, and ${\sf NC}$ formulations and introduce a new poset on Cambrian intervals to organize interval structure. The core ideas—a greedy chain construction, a local reordering lemma, and a combinatorial comparison criterion—yield a workable framework for studying $m$-Cambrian lattices and their intervals, with potential connections to $m$-Tamari and related interval enumeration. Overall, the work delivers a concrete, implementable method for working with $m$-Cambrian lattices and opens avenues for further structural and enumerative investigations in Coxeter combinatorics.

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 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. This work has been accepted as an extended abstract for the FPSAC 2025 conference. A long version of this work will be available later.

Figures (2)

• Figure 1: Schematic representation of the comparability condition $\leq_{(m)}$ on $m$-non-crossing partitions giving an order isomorphic to the $m$-Cambrian lattice.
• Figure 2: Three versions of $\operatorname{Camb}({\mathfrak{S_3}, {\sf s} {\sf t}})$. Here, ${\sf s} = (12)$, ${\sf t} = (23)$, ${\sf u} = (13)$.

Theorems & Definitions (28)

• Proposition 2.1
• Proposition 2.2: stump_cataland_2018
• Example 3.1
• Proposition 4.1
• Lemma 4.2
• Lemma 4.3
• proof
• Corollary 4.4
• proof
• Lemma 4.5: Local reordering lemma