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) 30 May 2019 On this day in 1814 Eugène Catalan was born. A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 1/5?

Background Facial Weak Order Properties 5 4 3 1 The facial weak order in hyperplane arrangements 2 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) 30 May 2019 On this day in 1814 Eugène Catalan was born. A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 1/5?

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 is codim 1 subspace of V with normal e H . Example A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 2/10?

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 Not central Central Central Not essential Not essential Essential A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 ∼ π /10?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Arrangements A - connected components of V without A . Regions R A - intersections of closures of some regions. Faces F 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 30 May 2019 4/10?

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Poset of regions A - some fixed region Base region B ∈ R Separation set for R ∈ R A 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 5/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Poset of regions A - some fixed region Base region B ∈ R Separation set for R ∈ R A 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 5/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Poset of regions A - some fixed region Base region B ∈ R Separation set for R ∈ R A 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 5/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Poset of regions A - some fixed region Base region B ∈ R Separation set for R ∈ R A 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 } ∅ 5/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019

Background Hyperplane Arrangements Facial Weak Order Poset of Regions Properties Motivation History and Background - Poset of regions A - some fixed region Base region B ∈ R 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 1 H 3 A { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } ∅ 5/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 A simplicial. Example Simplicial Not simplicial ∼ τ /10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 PR( A , B ) is a lattice for any B ∈ R A . If PR( A , B ) is a lattice then B is simplicial. Example 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 PR( A , B ) is a lattice for any B ∈ R A . If PR( A , B ) is a lattice then B is simplicial. Example 7/10 2 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 (aka Braid arrangement). 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. 8/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 (aka Braid arrangement). 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? 8/10 10 A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 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 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 ] 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 ] A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 9/ Ack( 100 , 100 )

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 A . Definition The facial weak order is the order FW( A , B ) on F A where for A : F , G ∈ F F ≤ G ⇔ m F ≤ PR m G and M F ≤ PR M G A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 10/ Ack( 100 , 100 )

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 ] A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 11/ Ack( 100 , 100 )

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 ] A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 12/ Ack( 100 , 100 )

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 ] A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 12/ Ack( 100 , 100 )

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 ] A. Dermenjian (UQÀM) The facial weak order in hyperplane arrangements 30 May 2019 12/ Ack( 100 , 100 )