Background Facial Weak Order The Process Extra Extra L ’ordre faible facial et tout son gloire Aram Dermenjian Présentation préntée comme exigence partielle du doctorat en mathématiques. Université du Québec à Montréal 30 août 2019 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 1/2?
Background Facial Weak Order The Process Extra Extra Outline How to arrange hyperplanes. The facial weak order in all its glory. The path of least resistance. What else? A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 2/5?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions How to arrange hyperplanes A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 ∼ π /10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A basic human problem A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 4/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A basic human problem A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 4/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A basic human problem A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 4/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A basic human problem A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 4/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions What is a hyperplane? ( V , �· , ·� ) - n -dim real Euclidean vector space. A hyperplane H is codim 1 subspace of V with normal e H . Example A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 5/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions Arranging hyperplanes A hyperplane arrangement is A = { H 1 , H 2 , . . . , H k } . A is central if { 0 } ⊆ � A . Central A is essential if { 0 } = � A . Example Not central Central Central Not essential Essential A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 ∼ τ /10 ?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions In terms of food? Central essential hyperplane arrangement A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019 7/10?
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions Exploding arrangements Regions R A - closures of connected components of V without A . Faces F A - intersections of some regions. H 3 H 1 F 3 F 2 R 3 R 4 R 2 − e 2 − e 3 − e 1 F 4 F 1 H 2 e 1 e 3 e 2 R 5 R 1 R 0 F 5 F 0 8/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional order Base region B ∈ R A - some fixed region Separation set for R ∈ R A S ( R ) := { H ∈ A | H separates R from B } H 3 H 1 R 3 R 4 R 2 H 2 R 5 R 1 B 9/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional order Base region B ∈ R A - some fixed region Separation set for R ∈ R A S ( R ) := { H ∈ A | H separates R from B } H 3 H 1 R 3 R 4 R 2 H 2 R 5 R 1 B 9/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional order Base region B ∈ R A - some fixed region Separation set for R ∈ R A S ( R ) := { H ∈ A | H separates R from B } H 3 H 1 R 3 { H 1 , H 2 } R 4 H 2 R 5 R 1 B 9/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional order Base region B ∈ R A - some fixed region Separation set for R ∈ R A S ( R ) := { H ∈ A | H separates R from B } H 3 H 1 A { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } ∅ 9/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional order Base region B ∈ R A - some fixed region Separation set for R ∈ R A S ( R ) := { H ∈ A | H separates R from B } Poset of Regions PR( A , B ) where R ≤ PR R ′ ⇔ S ( R ) ⊆ S ( R ′ ) H 3 H 1 A { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } ∅ 9/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions Ordering all the things 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 . . . . . . 8 12 . . . . . . 6 9 10 4 . . . . . . 2 3 5 7 1 10/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions Simply simplicial arrangements A region R is simplicial if normal vectors for boundary hyperplanes are linearly independent. A is simplicial if all R A simplicial. Example Simplicial Not simplicial 11/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional lattice 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. Example 12/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Background Facial Weak Order The Process Extra Extra Hyperplane Arrangements Poset of Regions A regional lattice 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. Example 12/10 2 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! Facial weak order in all its glory 13/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! Facial intervals Proposition (Björner, Las Vergas, Sturmfels, White, Ziegler ’93) Let A be central with base region B. For every F ∈ F A there is a unique interval [ m F , M F ] in PR( A , B ) such that [ m F , M F ] = { R ∈ R A | F ⊆ R } H 1 H 3 [ R 4 , R 3 ] [ R 2 , R 3 ] [ R 3 , R 3 ] R 3 F 3 F 2 R 3 [ R 4 , R 4 ] [ R 2 , R 2 ] R 4 R 2 R 4 R 2 [ R 5 , R 4 ] [ R 1 , R 2 ] F 4 F 1 0 [ B , R 3 ] H 2 R 5 R 1 R 5 R 1 [ R 5 , R 5 ] [ R 1 , R 1 ] B F 5 F 0 [ B , B ] B [ B , R 5 ] [ B , R 1 ] 14/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! Facial weak order (!!!) Let A be a central hyperplane arrangement and B a base region in R A . Definition The facial weak order is the order FW( A , B ) on F A where for F , G ∈ F A : F ≤ G ⇔ m F ≤ PR m G and M F ≤ PR M G M G m G M F m F 15/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! A first example R 3 [ R 4 , R 3 ] [ R 2 , R 3 ] R 4 R 2 [ R 3 , R 3 ] R 5 R 1 [ R 4 , R 4 ] [ R 2 , R 2 ] B [ B , R 3 ] [ R 5 , R 4 ] [ R 1 , R 2 ] [ R 5 , R 5 ] [ R 1 , R 1 ] [ B , B ] [ B , R 5 ] [ B , R 1 ] 16/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! A first example R 3 [ R 3 , R 3 ] [ R 4 , R 3 ] [ R 2 , R 3 ] R 4 R 2 [ R 4 , R 4 ] [ R 2 , R 2 ] R 5 R 1 B [ R 5 , R 4 ] [ B , R 3 ] [ R 1 , R 2 ] [ R 5 , R 5 ] [ R 1 , R 1 ] [ B , R 5 ] [ B , R 1 ] [ B , B ] 17/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! A first example R 3 [ R 3 , R 3 ] [ R 4 , R 3 ] [ R 2 , R 3 ] R 4 R 2 [ R 4 , R 4 ] [ R 2 , R 2 ] R 5 R 1 B [ R 5 , R 4 ] [ B , R 3 ] [ R 1 , R 2 ] [ R 5 , R 5 ] [ R 1 , R 1 ] [ B , R 5 ] [ B , R 1 ] [ B , B ] 17/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
Facial Intervals Lattice Background Facial Weak Order The Process Extra Extra All the definitions! A first example R 3 [ R 3 , R 3 ] [ R 4 , R 3 ] [ R 2 , R 3 ] R 4 R 2 [ R 4 , R 4 ] [ R 2 , R 2 ] R 5 R 1 B [ R 5 , R 4 ] [ B , R 3 ] [ R 1 , R 2 ] [ R 5 , R 5 ] [ R 1 , R 1 ] [ B , R 5 ] [ B , R 1 ] [ B , B ] 17/10 10 A. Dermenjian (UQAM) The facial weak order in all its glory 30 Aug 2019
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