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The facial weak order in hyperplane arrangements Aram Dermenjian (York Uni) , Christophe Hohlweg (LaCIM), Thomas McConville (UNC), Vincent Pilaud (LIX) Welcome! Thanks for coming to my poster talk! You can either go through the slides like “normal”, or jump around using the links in green (ex: Go to directory) or in the bottom-right corner of every slide. If you have any questions, don’t hesitate to ask Aram! Start with the directory Start with the main result! Link to directory arXiv: 1910.03511 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 Come back at any time Properties 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 e H . A hyperplane arrangement is A = { H 1 , H 2 , . . . , H k } . A is central if { 0 } ⊆ � A . H 1 H 3 Central A is essential if { 0 } = � A . H 2 e 1 e 3 e 2 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. H 1 H 3 F 3 F 2 R 3 R 4 R 2 F 4 F 1 H 2 e 1 e 3 R 5 R 1 e 2 R 0 F 5 F 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) 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 } H 1 H 3 The poset of regions PR( A , B ) is the A set of regions ordered by inclusion: R ≤ PR R ′ ⇔ S ( R ) ⊆ S ( R ′ ) { H 2 , H 3 } { H 1 , H 2 } H 2 { H 3 } { H 1 } S ( B ) = ∅ 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 R n 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 ) : � � [ m F , M F ] = R ∈ R A | F ⊆ R [ R 4 , R 3 ] [ R 2 , R 3 ] [ R 3 , R 3 ] R 3 [ R 4 , R 4 ] [ R 2 , R 2 ] R 4 R 2 [ R 5 , R 4 ] [ R 1 , R 2 ] [ B , R 3 ] R 5 R 1 [ R 5 , R 5 ] [ R 1 , R 1 ] B [ B , B ] [ B , R 5 ] [ B , R 1 ] 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. H 1 H 3 ( − , − , − ) ( 0 , − , − ) ( − , − , 0 ) (+ , − , − ) ( − , − , +) (+ , 0 , − ) ( − , 0 , +) H 2 e 1 e 3 e 2 (+ , + , − ) ( − , + , +) (+ , + , 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, [ m F , M F ] 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 Definition Definition F ≤ FW G F ≤ L G If | dim( F ) − dim( G ) | = 1 and ⇔ ⇔ 1. F ⊆ G , M F = M G , or m F ≤ PR m G F ( H ) ≥ G ( H ) 2. G ⊆ F , m F = m G . M F ≤ PR M G ( ∀ H ∈ A ) then F < · G . Theorem (Dermenjian, Hohlweg, McConville, Pilaud ’19+) ( F ≤ FW G ) ⇔ ( F = F 1 < · . . . < · F n = 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 of regions PR( A , B ) is a lattice then the facial weak order FW( A , B ) is a lattice. B 3 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 P op where x ≤ P y iff y ≤ P op x . Self-dual if P ∼ = P op . 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 M F is join-irreducible in PR( A , B ) and codim( F ) ∈ { 0 , 1 } . The Möbius function for X ≤ Y is given by: � ( − 1 ) rk( X )+rk( Y ) if X ≤ Z ≤ Y and Z = X − Z ∩ Y µ ( X , Y ) = 0 otherwise FPSAC 2020 Slides can be found at: dermenjian.com Back to: directory, results