Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Separating trees and simple congruences of the weak order

Emily Barnard, Jean-Christophe Novelli, Vincent Pilaud

TL;DR

This work studies simple congruences of the weak order on ${\mathfrak{S}}_n$, proving an elementary combinatorial characterization via forbidden up and down arcs and introducing separating trees as a precise model for the vertices of the corresponding quotientopes. It further extends to Schröder separating trees to capture all faces of the quotientope, providing a complete combinatorial description of the polytope's facial structure. An insertion-map framework ties the posets to separating trees, and the authors develop an algebraic structure on decorated separating trees while noting the absence of a Hopf coproduct; connections to quiver representation theory are established through torsion-class lattices and $g$-vector fans of certain quiver algebras. The results unify classical objects (Tamari, Cambrian lattices, permutrees) within a broader combinatorial-quiver framework, offering explicit tools to analyze simple congruences and their geometric realizations.

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.

Separating trees and simple congruences of the weak order

TL;DR

This work studies simple congruences of the weak order on , proving an elementary combinatorial characterization via forbidden up and down arcs and introducing separating trees as a precise model for the vertices of the corresponding quotientopes. It further extends to Schröder separating trees to capture all faces of the quotientope, providing a complete combinatorial description of the polytope's facial structure. An insertion-map framework ties the posets to separating trees, and the authors develop an algebraic structure on decorated separating trees while noting the absence of a Hopf coproduct; connections to quiver representation theory are established through torsion-class lattices and -vector fans of certain quiver algebras. The results unify classical objects (Tamari, Cambrian lattices, permutrees) within a broader combinatorial-quiver framework, offering explicit tools to analyze simple congruences and their geometric realizations.

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.

Paper Structure

This paper contains 17 sections, 28 theorems, 9 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Lattice congruences of the weak order
  3. Weak order, braid fan, permutahedron
  4. Noncrossing arc diagrams
  5. Lattice congruences and arc ideals
  6. Poset insertion map
  7. Quotient fans and quotientopes
  8. Relevant congruences
  9. Simple congruences
  10. Separating trees and posets of simple congruences
  11. Separating trees
  12. Posets of simple congruences
  13. Schröder separating trees and preposets of simple congruences
  14. Schröder separating trees
  15. Preposets of simple congruences
  16. ...and 2 more sections

Key Result

Theorem 2.1

The maps $\arcDiagramUp[](\sigma)$ and $\arcDiagramDown[](\sigma)$ are bijections from the permutations of ${\mathfrak{S}}_n$ to the noncrossing arc diagrams on ${\mathcal{A}}_n$.

Figures (12)

  • Figure 1: The Hasse diagram of the weak order on ${\mathfrak{S}}_4$ (left) can be seen as the dual graph of the braid fan $\braidFan[4]$ (middle) or as the graph of the permutahedron $\Perm[4]$ (right). PilaudSantos-quotientopes
  • Figure 2: The noncrossing arc diagrams $\arcDiagramDown[](\sigma)$ (bottom) and $\arcDiagramUp[](\sigma)$ (top) for the permutations $\sigma = 2537146$, $2531746$, $2513746$, and $2513476$. Pilaud-arcDiagramAlgebra
  • Figure 3: The subarc relation among arcs (left) and the subarc poset for $n = 4$ (right). The red arc $(i,j,A,B)$ is a subarc of the blue arc $(i',j',A',B')$. PilaudSantos-quotientopes
  • Figure 4: The weak order on ${\mathfrak{S}}_4$ (left), the sylvester congruence $\equiv_\textrm{sylv}$ (middle), and the Tamari lattice (right). PilaudSantos-quotientopes
  • Figure 5: The poset $\preccurlyeq_\sigma^\equiv$ (middle row) and the noncrossing arc diagrams $\arcDiagramUp(\sigma)$ (top row) and $\arcDiagramDown(\sigma)$ (bottom row) corresponding to the $\equiv_{\mathcal{A}}$-congruence class of the permutation $\sigma = 2537146$ for different arc ideals (represented in grey on the top and bottom rows).
  • ...and 7 more figures

Theorems & Definitions (87)

  • Theorem 2.1: Reading-arcDiagrams
  • Proposition 2.2
  • Theorem 2.3: Reading-arcDiagrams
  • Example 2.4
  • Proposition 2.5: Pilaud-arcDiagramAlgebra
  • Corollary 2.6
  • Example 2.7
  • Theorem 2.8: Reading-HopfAlgebras
  • Theorem 2.9: PilaudSantos-quotientopesPadrolPilaudRitter
  • Proposition 2.10: Reading-HopfAlgebrasPilaudSantos-quotientopesPadrolPilaudRitter
  • ...and 77 more