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Noncrossing arc diagrams of type B

Emily Barnard, Nathan Reading, Ashley M. Tharp

TL;DR

The paper develops two type-$B$ arc-diagram models for signed permutations—centrally symmetric diagrams and their orbifold quotients—and uses them to study lattice quotients of the weak order. By translating joins-irreducible elements and forcing into arc substructures, it extends the type-$A$ arc framework to type $B$, enabling concise, geometric descriptions of congruences, quotients, and Cambrian-type lattices. It provides explicit arc-based descriptions of parabolic, Cambrian, biCambrian, and linear biCambrian congruences, and shows how surjective homomorphisms from $B_n$ to $S_{n+1}$ arise from contracting specific join-irreducibles, with multiple equivalent models (centrally symmetric, orbifold) yielding streamlined proofs. The orbifold and symmetric viewpoints also clarify how canonical join representations behave under quotients and offer new proofs and simplifications for known congruence classifications in type $B$, with potential connections to quotientopes and lattice-theoretic realizations of quotients.

Abstract

Noncrossing arc diagrams are combinatorial models for permutations that encode information about lattice congruences of the weak order and about the associated discrete geometry. In this paper, we consider two related, analogous models for signed permutations. One model features centrally symmetric noncrossing arc diagrams, while the other features their quotients modulo the central symmetry. We demonstrate the utility of the models by applying them to various questions about lattice quotients of the weak order.

Noncrossing arc diagrams of type B

TL;DR

The paper develops two type- arc-diagram models for signed permutations—centrally symmetric diagrams and their orbifold quotients—and uses them to study lattice quotients of the weak order. By translating joins-irreducible elements and forcing into arc substructures, it extends the type- arc framework to type , enabling concise, geometric descriptions of congruences, quotients, and Cambrian-type lattices. It provides explicit arc-based descriptions of parabolic, Cambrian, biCambrian, and linear biCambrian congruences, and shows how surjective homomorphisms from to arise from contracting specific join-irreducibles, with multiple equivalent models (centrally symmetric, orbifold) yielding streamlined proofs. The orbifold and symmetric viewpoints also clarify how canonical join representations behave under quotients and offer new proofs and simplifications for known congruence classifications in type , with potential connections to quotientopes and lattice-theoretic realizations of quotients.

Abstract

Noncrossing arc diagrams are combinatorial models for permutations that encode information about lattice congruences of the weak order and about the associated discrete geometry. In this paper, we consider two related, analogous models for signed permutations. One model features centrally symmetric noncrossing arc diagrams, while the other features their quotients modulo the central symmetry. We demonstrate the utility of the models by applying them to various questions about lattice quotients of the weak order.

Paper Structure

This paper contains 15 sections, 45 theorems, 24 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices and congruences
  4. Coxeter groups and the weak order
  5. Shards, congruences, and forcing
  6. Noncrossing arc diagrams of type A
  7. Centrally symmetric noncrossing arc diagrams
  8. Orbifold noncrossing arc diagrams
  9. Congruence examples
  10. Parabolic congruences
  11. Surjective homomorphisms from $B_n$ to $S_{n+1}$
  12. Cambrian congruences
  13. Bipartite biCambrian congruences
  14. Linear biCambrian congruences
  15. Symmetric congruences of the weak order on $S_{2n}$

Key Result

Theorem 2.1

Suppose $L$ is a finite join-semidistributive lattice. The canonical join complex of $L$ is flag if and only if $L$ is semidistributive.

Figures (17)

  • Figure 1: The map $\delta$ applied to $6437125$
  • Figure 2: The map from type-A noncrossing arc diagrams to permutations
  • Figure 3: Centrally symmetric noncrossing arc diagrams for $B_2$
  • Figure 4: The map $\delta$ applied to $\pi=(-4)352(-1)$
  • Figure 5: Subarcs and subarc pairs
  • ...and 12 more figures

Theorems & Definitions (64)

  • Theorem 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Proposition 2.8
  • proof
  • Theorem 2.9
  • ...and 54 more