Séminaire Lotharingien de Combinatoire, 95B.36 (2026), 12 pp.
Nathan Reading and Tanvi Thummar
m is for meet-congruence
Abstract.
Lattice congruences and quotients of the weak order on a finite Coxeter group are a rich source of combinatorial insight.
Recently, Stump, Thomas, and Williams defined the m-eralized weak order associated to a finite Coxeter group, as a way of constructing m-eralized Cambrian lattices.
The case m = 1 recovers the usual weak order and Cambrian lattices.
Lattice congruences and quotients are less useful in the case m
> 1, in a sense that we make precise.
We show, by several explicit constructions, that meet-congruences and meet-quotients play a crucial role in the combinatorics of the m-weak order.
Most notably, we show that each m-eralized Cambrian lattice is a meet-quotient of the m-eralized weak order and construct a meet-congruence that relates to an m-version of the descent map.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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