Separating trees and simple congruences of the weak order
arXiv:2503.15053 · doi:10.1016/j.ejc.2026.104375
Abstract
A congruence of the weak order is simple if its quotientope is a simple polytope. We provide an alternative elementary proof of the characterization of the simple congruences in terms of forbidden up and down arcs. For this, we provide a combinatorial description of the vertices of the corresponding quotientopes in terms of separating trees. This also yields a combinatorial description of all faces of the corresponding quotientopes. We finally explore algebraic aspects of separating trees, in particular their connections with quiver representation theory.
24 pages, 12 figures; Version 2: minor corrections, published version