[PPT] - The facial weak order in hyperplane arrangements Aram Dermenjian 1,3 PowerPoint Presentation

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Background Facial Weak Order Properties

The facial weak order in hyperplane arrangements

Aram Dermenjian1,3

Christophe Hohlweg1, Thomas McConville2 and Vincent Pilaud3 1Université du Québec à Montréal (UQAM) 2Mathematical Sciences Research Institute (MSRI) 3École Polytechnique (LIX)

22 April 2019

On this day in 1811 Otto Hesse was born.

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 1/5?

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Background Facial Weak Order Properties

Outline

Arranging hyperplanes. The facial weak order and its 1, 2, 3, 4 (!) definitions. Yeah, but is it a lattice? Some other properties.

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 2/10?

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Hyperplanes

(V, ·, ·) - n-dim real Euclidean vector space. A hyperplane Hi is codim 1 subspace of V with normal ei. Example

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The facial weak order in hyperplane arrangements 22 Apr 2019 ∼π/10?

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Arrangements

A hyperplane arrangement is A = {H1, H2, . . . , Hk}. A is central if {0} ⊆ A. Central A is essential if {0} = A. Example

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The facial weak order in hyperplane arrangements 22 Apr 2019 4/10?

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Arrangements

Regions R - connected components of V without A. Faces FA - intersections of closures of some regions.

−e1 −e2 −e3 e1 e2 e3

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

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The facial weak order in hyperplane arrangements 22 Apr 2019 5/10?

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - (Partial) Orders

Lattice - poset where every two elements have a meet (greatest lower bound) and join (least upper bound). Example The lattice (N, |) where a ≤ b ⇔ a | b. meet - greatest common divisor join - least common multiple 1 2 3 4 5 6 7 8 9 10 12 . . . . . . . . . . . . . . . . . .

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 ∼τ/10?

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Base region B ∈ R - some fixed region Separation set for R ∈ R S(R) := {H ∈ A | H separates R from B}

H3 H1 H2 B R1 R3 R4 R5 R2

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 7/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Base region B ∈ R - some fixed region Separation set for R ∈ R S(R) := {H ∈ A | H separates R from B}

H3 H1 H2 B R1 R3 R4 R5 R2

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 7/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Base region B ∈ R - some fixed region Separation set for R ∈ R S(R) := {H ∈ A | H separates R from B}

H3 H1 H2 B R1 R3 R4 R5 {H1, H2}

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 7/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Base region B ∈ R - some fixed region Separation set for R ∈ R S(R) := {H ∈ A | H separates R from B}

H3 H1 H2 {H1, H2} ∅ {H1} A {H2, H3} {H3}

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 7/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Base region B ∈ R - some fixed region Separation set for R ∈ R S(R) := {H ∈ A | H separates R from B} Poset of Regions (R, B, ≤A) where R ≤A R′ ⇔ S(R) ⊆ S(R′)

H3 H1 H2 {H1, H2} ∅ {H1} A {H2, H3} {H3}

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 7/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

A region R is simplicial if normal vectors for boundary hyperplanes are linearly independent. A is simplicial if all R simplicial. Example

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The facial weak order in hyperplane arrangements 22 Apr 2019 8/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Theorem (Björner, Edelman, Zieglar ’90) If A is simplicial then (R, B, ≤A) is a lattice for any B ∈ R. If (R, B, ≤A) is a lattice then B is simplicial. Example

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The facial weak order in hyperplane arrangements 22 Apr 2019 9/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

History and Background - Poset of regions

Theorem (Björner, Edelman, Zieglar ’90) If A is simplicial then (R, B, ≤A) is a lattice for any B ∈ R. If (R, B, ≤A) is a lattice then B is simplicial. Example

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 9/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

Coxeter Arrangements

Example A Coxeter arrangement is the hyerplane arrangement associated to a Coxeter group. Coxeter Groups Hyperplane Arrangements Reflecting hyperplanes ↔ Hyperplane arrangement Root system ↔ Normals to hyperplanes Inversion sets ↔ Seperation sets Weak order ↔ Poset of regions

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The facial weak order in hyperplane arrangements 22 Apr 2019 10/102

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

Motivation

In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order of Coxeter groups to an order on all the faces of its associated arrangement for type A. In 2006, Palacios and Ronco extended this new order to Coxeter groups of all types using cover relations. In 2016, D, Hohlweg and Pilaud showed this extension has a global equivalent and produces a lattice in Coxeter arrangements.

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 11/1010

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Background Facial Weak Order Properties Hyperplane Arrangements Poset of Regions Motivation

Motivation

In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order of Coxeter groups to an order on all the faces of its associated arrangement for type A. In 2006, Palacios and Ronco extended this new order to Coxeter groups of all types using cover relations. In 2016, D, Hohlweg and Pilaud showed this extension has a global equivalent and produces a lattice in Coxeter arrangements. Questions: Can we extend this to hyperplane arrangements? Can we find both local and global definitions? When do we actually get a lattice?

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 11/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Intervals

Proposition (Björner, Las Vergas, Sturmfels, White, Ziegler ’93) Let A be central with base region B. For every F ∈ FA there is a unique interval [mF, MF] in (R, B, ≤A) such that [mF, MF] =

R ∈ R | F ⊆ R
H3

H1 H2 F0 F1 F2 F3 F4 F5 B R1 R2 R3 R4 R5 [B, R1] [R2, R3] [R4, R3] [B, R5] [R1, R2] [R5, R4] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3]

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 12/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order

Let A be a central hyperplane arrangement and B a base region in R. Definition The facial weak order is the order FW(A, B) on FA where for F, G ∈ F: F ≤ G ⇔ mF ≤A mG and MF ≤A MG

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The facial weak order in hyperplane arrangements 22 Apr 2019 13/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 14/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial Weak Order - Example

B R1 R2 R3 R4 R5

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 15/1010

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Cover Relations

Proposition (D., Hohlweg, McConville, Pilaud, ’18+) For F, G ∈ FA if

1. F ≤ G in FW(A, B)
2. |dim(F) − dim(G)| = 1
3. F ⊆ G or G ⊆ F

then F < · G.

F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 16/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Covectors

covector - a vector in {−, 0, +}A with signs relative to hyperplanes. L ⊆ {−, 0, +}A - set of covectors Example F4 ↔ (+, 0, −) F4(H1) = +; F4(H2) = 0; F4(H3) = −

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 17/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Covectors

covector - a vector in {−, 0, +}A with signs relative to hyperplanes. L ⊆ {−, 0, +}A - set of covectors Example F4 ↔ (+, 0, −) F4(H1) = +; F4(H2) = 0; F4(H3) = −

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 17/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Covector operations

For X, Y ∈ L ⊆ {−, 0, +}A Composition: (X ◦ Y)(H) =

Y(H)

if X(H) = 0 X(H)

therwise

Reorientation: (X−Y) (H) =

−X(H)

if Y(H) = 0 X(H)

therwise

⋆ For F ∈ FA, [mF, MF] = [F ◦ B, F ◦ −B] Example Let A = {H1, H2, H3, H4, H5}. X = (−, 0, +, +, 0) Y = (0, 0, −, 0, +) Then X ◦ Y = (−, 0, +, +, +) X−Y = (+, 0, +, −, 0)

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The facial weak order in hyperplane arrangements 22 Apr 2019 18/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Covector Definition

Definition For X, Y ∈ L: X ≤L Y ⇔ Y(H) ≤ X(H) ∀H with − < 0 < +

−e1 −e2 −e3 e1 e2 e3 H3 H1 H2 (0, +, +) (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0)

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 19/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotopes

Zonotope ZA is the convex polytope: ZA :=

  v ∈ V | v =

k

i=1

λiei, such that |λi| ≤ 1 for all i

  

Theorem (Edelman ’84, McMullen ’71) There is a bijection between FA and the nonempty faces of ZA given by the map τ(F) =

  v ∈ V | v =

F(Hi)=0

λiei +

F(Hj)=0

µjej

  

where |λi| ≤ 1 for all i and µj = F(Hj)

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 20/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

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

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

H3 H1 H2 F0 F2 F3 F4 F5 τ(F1) B R1 R2 R3 R4 R5

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Zonotope - Construction example

−e1 −e2 −e3 e1 e2 e3

H3 H1 H2 τ(F1) τ(F2) τ(F3) τ(F4) τ(F5) τ(F0) τ(B) τ(R1) τ(R2) τ(R3) τ(R4) τ(R5)

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 21/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Root inversion sets

roots ΦA := {±e1, ±e2, . . . , ±ek} root inversion set R(F) := {e ∈ ΦA | x, e ≤ 0 for some x ∈ F}. R(R4) R(R3) R(R5) R(R2) R(B) R(R1) R(F4) R(F3) R(F5) R(F2) R(F0) R(F1) R({0}) τ(R2) τ(R3) τ(R4) τ(R5) τ(B) τ(R1) τ(F5) τ(F0) τ(F1) τ(F2) τ(F3) τ(F4) {0}

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 22/Ack(100, 100)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalent definitions

Theorem (D., Hohlweg, McConville, Pilaud ’18+) For F, G ∈ FA the following are equivalent: mF ≤A mG and MF ≤A MG in poset of regions (R, B, ≤A). There exists a chain of covers in FW(A, B) such that F = F1 < · F2 < · · · · < · Fn = G F ≤L G in terms of covectors (G(H) ≤ F(H) ∀H ∈ A) R(F)\R(G) ⊆ Φ−

A and R(G)\R(F) ⊆ Φ+ A.

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 23/a lot

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

R(F)\R(G) ⊆ Φ−

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

SLIDE 45

Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

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SLIDE 46

Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

SLIDE 47

Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG mF ≤A mG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, R1] [R4, R3] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

The facial weak order in hyperplane arrangements 22 Apr 2019 24/a lot

SLIDE 48

Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Equivalence for type A2 Coxeter arrangement

mF ≤A mG MF ≤A MG MF ≤A MG B R1 R2 R3 R4 R5 [R1, R2] [R2, R3] [R4, R3] [R5, R4] [B, R5] [B, R1] [B, R1] [R4, R3] [B, B] [R1, R1] [R2, R2] [R3, R3] [R4, R4] [R5, R5] [B, R3] F ≤ G |dim F − dim G| = 1 F ⊆ G or G ⊆ F F0 F1 F2 F3 F4 F5 R0 R1 R2 R3 R4 R5 F1 F2 F3 F4 F5 F0 B R1 R2 R3 R4 R5 G(H) ≤ F(H) ∀H ∈ A (−, 0, +) (−, −, 0) (0, −, −) (+, 0, −) (+, +, 0) (0, +, +) (+, +, +) (−, +, +) (−, −, +) (−, −, −) (+, −, −) (+, +, −) (0, 0, 0) R(F)\R(G) ⊆ Φ−

A

R(G)\R(F) ⊆ Φ+

A

e1 e2 e3 −e1 −e2 −e3

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Facial weak order lattice

Theorem (D., Hohlweg, McConville, Pilaud ’18+) The facial weak order FW(A, B) is a lattice when A is simplicial. Corollary (D., Hohlweg, McConville, Pilaud ’18+) The lattice of regions is a sublattice of the facial weak order lattice when A is simplicial.

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Lattice proof - Joins

Proof uses two key components : Lemma (Björner, Edelman, Zieglar ’90) 1: If L is a finite, bounded poset such that x ∨ y exists whenever x and y both cover some z ∈ L, then L is a lattice. 2: Cover relation: Z < · X iff Z ≤ X, |dim X − dim Z| = 1 and X ⊆ Z or Z ⊆ X. Then Z < · X and Z < · Y gives three cases:

1. X ∪ Y ⊆ Z and dim X = dim Y = dim Z − 1,
2. Z ⊆ X ∩ Y and dim X = dim Y = dim Z + 1, and
3. X ⊆ Z ⊆ Y and dim X = dim Z − 1 = dim Y − 2.
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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ∪ Y ⊆ Z and dim X = dim Y = dim Z − 1

H3 H1 H2 F0 = X F1 F2 F3 F4 F5 = Y B = Z R1 R2 R3 R4 R5

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ∪ Y ⊆ Z and dim X = dim Y = dim Z − 1

H3 H1 H2 F0 = X F1 F2 F3 F4 F5 = Y B = Z R1 R2 R3 R4 R5

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Z ⊆ X ∩ Y and dim X = dim Y = dim Z + 1

H3 H1 H2 F0 F1 F2 = Y F3 = X F4 F5 B R1 R2 R3 R4 R5 0 = Z

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Z ⊆ X ∩ Y and dim X = dim Y = dim Z + 1

H3 H1 H2 F0 F1 F2 = Y F3 = X F4 F5 B R1 R2 R3 R4 R5 0 = Z

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ⊆ Z ⊆ Y and dim X = dim Z − 1 = dim Y − 2

H3 H1 H2 F0 F1 F2 F3 F4 F5 = Z B R1 R2 R3 R4 R5 = Y 0 = X

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ⊆ Z ⊆ Y and dim X = dim Z − 1 = dim Y − 2

H3 H1 H2 F0 F1 F2 F3 F4 F5 = Z B = Y−Z R1 R2 R3 R4 R5 = Y 0 = X

A. Dermenjian (UQÀM)

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SLIDE 57

Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ⊆ Z ⊆ Y and dim X = dim Z − 1 = dim Y − 2

H3 H1 H2 F0 F1 F2 F3 F4 F5 = Z B = Y−Z R1 R2 R3 R4 R5 = Y 0 = X

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

X ⊆ Z ⊆ Y and dim X = dim Z − 1 = dim Y − 2

H3 H1 H2 F0 F1 F2 F3 F4 F5 = Z B = Y−Z R1 R2 R3 R4 R5 = Y 0 = X

A. Dermenjian (UQÀM)

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Background Facial Weak Order Properties Facial Intervals All the definitions! Lattice

Example: B3 Coxeter arrangement

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Background Facial Weak Order Properties Properties Further Works

Properties of the facial weak order

The dual of a poset P is the poset Pop where x ≤ y in P iff y ≤ x in Pop. A poset is self-dual if P ∼ = Pop. A lattice is semi-distributive if x ∨ y = x ∨ z implies x ∨ y = x ∨ (y ∧ z) and similarly for the meets. Theorem (D., Hohlweg, McConville, Pilaud ’18+) The facial weak order FW(A, B) is self-dual. If furthermore, A is simplicial, FW(A, B) is a semi-distributive lattice.

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Background Facial Weak Order Properties Properties Further Works

Join-irreducible elements

An element is join-irreducible if and only if it covers exactly

ne element.

Proposition (D., Hohlweg, McConville, Pilaud ’18+) If A is simplicial and F a face with facial interval [mF, MF]. Then F is join-irreducible in FW(A, B) if and only if MF is join-irreducible in (R, B, ≤A) and codim(F) ∈ {0, 1}

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Background Facial Weak Order Properties Properties Further Works

Möbius function

Recall that the Möbius function is given by: µ(x, y) =

      

1 if x = y −

x≤z<y µ(x, z)

if x < y

therwise

Proposition (D., Hohlweg, McConville, Pilaud ’18+) Let X and Y be faces such that X ≤ Y and let Z = X ∩ Y. µ(X, Y) =

(−1)rk(X)+rk(Y)

if X ≤ Z ≤ Y and Z = X−Z ∩ Y

therwise
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Background Facial Weak Order Properties Properties Further Works

Further Works

Can we explicitly state the join/meet of two elements? When is the facial weak order congruence uniform? Can we generalize this to polytopes?

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Background Facial Weak Order Properties Properties Further Works

Thank you!

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