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The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

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arXiv: 1910.03511

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FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Directory of contents

Background: Hyperplane arrangements Regions and faces Poset of regions Lattice of regions Facial Weak Order: Facial intervals Covectors Facial weak order Our main results Properties

Come back at any time

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Hyperplane arrangements

Let (V, ·, ·) be an n-dim real Euclidean vector space. A hyperplane H is codim 1 subspace of V with normal eH. A hyperplane arrangement is A = {H1, H2, . . . , Hk}. A is central if {0} ⊆ A. Central A is essential if {0} = A.

e1 e2 e3

H3 H1 H2

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Regions and faces

Let A be an arrangement. Regions R

A - connected components of V without A.

Faces F

A - intersections of closures of some regions.

e1 e2 e3

H3 H1 H2 F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Poset of regions

Base region B - some fixed region in R

A.

Separation set for R ∈ R

A

S(R) := {H ∈ A | H separates R from B} The poset of regions PR(A, B) is the set of regions ordered by inclusion: R ≤PR R′ ⇔ S(R) ⊆ S(R′)

H3 H1 H2 {H1, H2} S(B) = ∅ {H1} A {H2, H3} {H3}

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Lattice of regions

An arrangement A in Rn is simplicial if every region is simplicial (i.e., has n boundary hyperplanes). Theorem (Björner, Edelman, Ziegler ’90) If A is simplicial then PR(A, B) is a lattice for any B ∈ R

A.

If PR(A, B) is a lattice then B is simplicial.

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Facial intervals

Proposition (Björner, Las Vergnas, Sturmfels, White, Ziegler ’93) For every F ∈ F

A there is a unique interval in PR(A, B):

[mF, MF] =

• R ∈ R

A | F ⊆ R

• B

R5 R1 R4 R2 R3 [B, R5] [R2, R3] [R4, R3] [B, R1] [R1, R2] [R5, R4] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3]

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Covectors

A covector of a face is a sign vector in {−, 0, +}A relative to hyperplanes.

e1 e2 e3

H3 H1 H2 (0, +, +) (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −)

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Facial weak order

Let PR(A, B) be the poset of regions, [mF, MF] be the facial interval of a face F and L be the set of covectors. The facial weak order, FW(A, B), is the partial order ≤FW on the set of faces (the left-hand definition). Let F, G by faces in F

A:

Definition F ≤FW G ⇔ mF ≤PR mG MF ≤PR MG Definition If |dim(F) − dim(G)| = 1 and

• 1. F ⊆ G, MF = MG, or
• 2. G ⊆ F, mF = mG.

then F < · G. Definition F ≤L G ⇔ F(H) ≥ G(H) (∀H ∈ A) Theorem (Dermenjian, Hohlweg, McConville, Pilaud ’19+) (F ≤FW G) ⇔ (F = F1 < · . . . < · Fn = G) ⇔ (F ≤L G)

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Main results

Theorem (Dermenjian, Hohlweg, McConville, Pilaud ’19+) Let A be an arrangement and fix a base region B. If the poset

• f regions PR(A, B) is a lattice then the facial weak order

FW(A, B) is a lattice. B3 Example: Properties of the facial weak order →

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results

The facial weak order in hyperplane arrangements

Aram Dermenjian (York Uni), Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX)

Properties of the facial weak order

• 1. Dual of a poset P is the poset Pop where x ≤P y iff y ≤Pop x. Self-dual if P ∼

= Pop.

• 2. A lattice is semi-distributive if x ∨ y = x ∨ z implies x ∨ y = x ∨ (y ∧ z) and similarly

for meets.

• 3. x ∈ P is join-irreducible if it covers exactly one element.

Theorem (Dermenjian, Hohlweg, McConville, Pilaud ’19+) Facial weak order is self-dual. If A is simplicial then the facial weak order is semi-distributive. If A is simplicial then F is join-irreducible if and only if MF is join-irreducible in PR(A, B) and codim(F) ∈ {0, 1}. The Möbius function for X ≤ Y is given by: µ(X, Y) =

• (−1)rk(X)+rk(Y)

if X ≤ Z ≤ Y and Z = X−Z ∩ Y

• therwise

FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results