Lattices From Graph Associahedra Emily Barnard Joint with Thomas - - PowerPoint PPT Presentation

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Lattices From Graph Associahedra Emily Barnard Joint with Thomas - - PowerPoint PPT Presentation

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Lattices From Graph Associahedra Emily Barnard Joint with Thomas McConville DePaul University July 1, 2019 My favorite posets Posets From Polytopes Definition Let P a polytope with vertex set V , and fix a linear function . Let L p P ,

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Lattices From Graph Associahedra

Emily Barnard

Joint with Thomas McConville

DePaul University

July 1, 2019

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My favorite posets

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Posets From Polytopes

Definition

Let P a polytope with vertex set V , and fix a linear function λ. Let LpP, λq denote the partial order on V obtained by taking the transitive and reflexive closure of x ď y when

  • rx, ys is an edge of P and
  • λpxq ď λpyq.
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Posets from Polytopes

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Posets from Polytopes

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Posets from (normal) fans

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Posets from (normal) fans

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Posets from (normal) fans

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Motivation

Properties of the weak order on Sn and the Tamari lattice

  • The Hasse diagram is (an orientation of) the one-skeleton of a

polytope.

  • Both posets are lattices.

Fact

  • The normal fan of the associahedron coarsens the normal fan
  • f the permutahedron.
  • Thus, there is a canonical surjection from Sn onto the Tamari

lattice Tn which we denote by Ψ.

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The Canonical Surjection

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The Canonical Surjection

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The Canonical Surjection

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The Canonical Surjection

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The Canonical Surjection

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Goal of the talk

Theorem [Reading]

The canonical surjection Ψ : Sn Ñ Tn is a lattice quotient. That is:

  • Ψpw _ w1q “ Ψpwq _ Ψpw1q
  • Ψpw ^ w1q “ Ψpwq ^ Ψpw1q

Set Up

Given a graph G, we will construct a graph associahedron PG, a polytope whose normal fan coarsens the normal fan of the

  • permutahedron. Then we will construct an analogous poset LG.

Question

For which G is the canonical surjection ΨG : Sn Ñ LG a lattice quotient?

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Notation

  • Write rns for the set t1, 2, . . . , nu.
  • G is a graph with vertex set rns.
  • Let ∆I denote the simplex with vertex set tei : i P I Ď rnsu.

Definition/Recall

Let P and Q be polytopes. The Minkowski Sum is the polytope P ` Q “ tx ` y : x P P and y P Qu. The normal fan of P is a coarsening of the normal fan of P ` Q.

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Graph Associahedra

Definition

A tube is a nonempty subset I of vertices such that the induced subgraph G|I is connected.

The Graph Assocciahedron

The Graph Associahedron PG is the Minkowski sum PG “ ÿ

I is a tube of G

∆I.

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Examples: The Complete Graph

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Examples: The Complete Graph

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Examples: The Complete Graph

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Examples: The Complete Graph

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Examples: The Complete Graph

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Examples

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Context: Topology and Geometry

The Bergman Complex

Let M be an oriented matroid. The Bergman complex BpMq and the positive Bergman complex B`pMq generalize the notions of a tropical variety and positive tropical variety to matroids.

Theorem[Ardila, Reiner, Williams]

Let Φ be a the root system associated to a (possibly infinite) Coxeter system pW , Sq and let Γ be the associated Coxeter

  • diagram. The positive Bergman complex B`pMΦq is dual to the

graph associahedron PΓ.

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The Poset LG

Definition

Fix λ “ pn, n ´ 1, . . . , 2, 1q. The poset LG is the partial order on the vertex set of PG obtained by taking the transitive and reflexive closure of x ď y when

  • rx, ys is an edge of PG and
  • λpxq ď λpyq.
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The poset LG

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The poset LG

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The canonical surjection

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The canonical surjection

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The canonical surjection

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The Canonical Surjection

Let ΨG denote the surjection from Sn to the poset LG.

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Recap: Main Question

Theorem [Reading]

Let G be the path graph, let LG be the associated poset. Then canonical surjection ΨG : Sn Ñ LG is a lattice quotient. That is:

  • ΨGpw _ w1q “ ΨGpwq _ ΨGpw1q
  • ΨGpw ^ w1q “ ΨGpwq ^ ΨGpw1q

Question

For which G is the canonical surjection ΨG a lattice quotient?

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Main Results

Definition

We say a graph G is filled if for each edge ti, ku in G, the edges ti, ju and tj, ku are also in G for all i ă j ă k.

Theorem [B., McConville]

The map ΨG is a lattice quotient if and only if G is filled.

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A filled graph

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Proof Sketch

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Hvala! Thank you!

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When is LG a lattice?

Definition

Two tubes I, J are said to be compatible if either

  • they are nested: I Ď J or J Ď I, or
  • they are separated: I Y J is not a tube.

A (maximal) tubing X of G is a (maximal) collection of pairwise compatible tubes.

Definition/Theorem

Each cover relation in LG is encoded by a flip X Ñ Y defined by:

  • Y “ XztIu Y tI 1u
  • topX pIq ă topYpI 1q
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When is LG a lattice?

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4