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arXiv · 2503.15053

Separating trees and simple congruences of the weak order

Emily Barnard·Jean-Christophe Novelli·Vincent Pilaud
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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.

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Emily Barnard, Jean-Christophe Novelli, Vincent Pilaud. 2025-03-19. Separating trees and simple congruences of the weak order. https://doi.org/10.1016/j.ejc.2026.104375

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