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

Background Facial Weak Order Properties The facial weak order in hyperplane arrangements Aram Dermenjian 1,3 Christophe Hohlweg 1 , Thomas McConville 2 and Vincent Pilaud 3 1 Université du Québec à Montréal (UQAM) 2 Mathematical Sciences Research Institute (MSRI) 3 École 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?

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?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Hyperplanes ( V , �· , ·� ) - n -dim real Euclidean vector space. A hyperplane H i is codim 1 subspace of V with normal e i . Example A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019 ∼ π /10?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Arrangements 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 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019 4/10?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Arrangements Regions R - connected components of V without A . Faces F A - intersections of closures of some regions. H 1 H 3 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 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019 5/10?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 . . . . . . 8 12 . . . . . . 4 6 9 10 . . . 2 3 5 7 . . . 1 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019 ∼ τ /10 ?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 } H 1 H 3 R 3 R 4 R 2 H 2 R 5 R 1 B 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 } H 1 H 3 R 3 R 4 R 2 H 2 R 5 R 1 B 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 } H 1 H 3 R 3 R 4 { H 1 , H 2 } H 2 R 5 R 1 B 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 } H 1 H 3 A { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } ∅ 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 ′ ) H 1 H 3 A { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } ∅ 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 8/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 9/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 9/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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 ↔ 10/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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. 11/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties 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? 11/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Facial Intervals Facial Weak Order All the definitions! Properties Lattice 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 ( R , B , ≤ A ) such that � � [ m F , M F ] = R ∈ R | F ⊆ R H 1 H 3 [ R 4 , R 3 ] [ R 2 , R 3 ] [ R 3 , R 3 ] F 3 F 2 R 3 [ R 4 , R 4 ] [ R 2 , R 2 ] R 4 R 2 F 4 0 F 1 [ B , R 3 ] H 2 [ R 5 , R 4 ] [ R 1 , R 2 ] R 5 R 1 [ R 5 , R 5 ] [ R 1 , R 1 ] B F 5 F 0 [ B , B ] [ B , R 5 ] [ B , R 1 ] 12/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Facial Intervals Facial Weak Order All the definitions! Properties 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 F A where for F , G ∈ F : F ≤ G ⇔ m F ≤ A m G and M F ≤ A M G 13/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Facial Intervals Facial Weak Order All the definitions! Properties Lattice Facial Weak Order - 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 ] 14/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019

Background Facial Intervals Facial Weak Order All the definitions! Properties Lattice Facial Weak Order - 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 ] 15/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 22 Apr 2019