Coherence of f -Monotone Paths on Zonotopes. Robert Edman May 15, 2015 1 / 30
An Analogy: The Secondary Polytope Definition (Polytope) A polytope is a convex hull of finitely many points in R d . Combinatorially a polytope can be defined by its face lattice. Definition (Polyhedral Subdivision) A polyhedral subdivision is a decomposition of P into subpolytopes. A subdivision is a triangulation when each subpolytope is a simplex. 2 / 30
Remark Subdivisions of P form a poset called the refinement poset of P. 3 / 30
Remark In this example, the refinement poset is the face lattice of a polytope. 4 / 30
◮ Some bad triangulations are not regular or are incoherent . ◮ Coherence is a linear inequality condition. ◮ Σ( P ) is an example of a Fiber Polytope . Theorem (GKZ) The refinement poset of all regular subdivisions of P is the face lattice of a polytope Σ( P ) . 5 / 30
Our Work: Monotone Paths v 5 ◮ Our version of triangulations are f-monotone edge paths of P . ◮ f must be generic , f v 4 non-constant on each edge of P . v 2 v 3 ◮ The refinement poset consists of cellular strings. v 1 Definition An f-monotone edge path is a path from the f-minimal vertex − z to the f-maximal vertex z along the edges of P. 6 / 30
Definition ◮ The vertices graph G 2 ( P , f ) is formed from all elements on the bottom level levels of the refinement poset. ◮ In this example every f-monotone path is coherent. 7 / 30
Question When does P have incoherent f-monotone paths? 8 / 30
Definition (Coherent) � R d � ∗ An f-monotone path γ is coherent if there exists g ∈ making γ the lower face of the polytope P = Conv { ( f ( p i ) , g ( p i )) } ⊂ R 2 . z 3 2 1 3 2 2 1 1 1 2 f 3 3 3 3 g − z z 2 1 1 2 1 2 3 1 2 3 f − z Remark The refinement poset of coherent cellular strings is the fiber polytope Σ( P , f ) . 9 / 30
4321 4231 3421 2431 3241 2341 4312 4213 3412 2413 3214 2314 f 4132 4123 3142 2143 3124 2134 2 3 1432 1423 4 1 1342 1243 1324 1234 Theorem (Billera & Sturmfels) Every f-monotone path of a cube is coherent. 10 / 30
Definition ◮ A zonotope is the image of the n-cube in R d under a projection A : C n → R d specified by a d × n matrix 12 3 + − −− 4 − − −− | | | A A = a 1 a 2 . . . a n | | | ◮ The zonotope Z ( A ) = � [ − a i , + a i ] is the Minkowski of the columns of A . 2 3 ◮ The vertices of Z ( A ) are sign 1 4 + − −− − − −− vectors 11 / 30
Proposition ◮ Every f-monotone path of Z ( A ) is of length n. ◮ The function f is generic when f ( a i ) > 0 for all i. ◮ The choice of f corresponds to the choice of a f-minimal vertex − z. ◮ But not all vertices are symmetric, so we will have to consider multiple options for z. ◮ The corank of Z is n − d. 12 / 30
z γ ( 4 ) γ ( 3 ) f g g γ ( 4 ) g γ ( 1 ) f γ ( 4 ) γ ( 2 ) f γ ( 1 ) g γ ( 2 ) g γ ( 3 ) γ ( 1 ) f γ ( 2 ) f γ ( 3 ) − z f Proposition � R d � ∗ so A f-monotone path γ is coherent if there exists a g ∈ that: g γ ( 1 ) < g γ ( 2 ) < . . . < g γ ( n ) f γ ( 1 ) f γ ( 2 ) f γ ( n ) 13 / 30
Corank 1 a 1 a 2 a 3 a 4 1 1 1 1 Z ( 4 , 3 ) = 1 2 3 4 1 4 9 16 − + ++ + + ++ Remark ◮ Every f-monotone path is coherent for − + ++ . ◮ + + ++ has an incoherent f-monotone path for every f. 14 / 30
Corank 2 (cyclic) a 1 a 2 a 3 a 4 a 5 1 1 1 1 1 Z ( 5 , 3 ) = 1 2 3 4 5 1 4 9 16 25 − + + + + − − + + + + + + + + Remark ◮ Has incoherent f-monotone path for every f. ◮ + + + + + is an important geometric counterexample. 15 / 30
Definition (Pointed hyperplane arrangement) The normal fan of the zonotope, is a hyperplane arrangement, � � a ⊥ 1 , . . . , a ⊥ A = . The choice of a chamber c of A n corresponds to the choice of f. a ⊥ ◮ Easy to draw under 3 a ⊥ 1 stereographic projection ◮ k -faces of Z ⇐ ⇒ d − k intersections of a ⊥ 2 X 2 , 3 hyperplanes. − − − − − ◮ L 2 ( A ) are the codimension a ⊥ 5 2 intersections of a ⊥ 4 hyperplanes. 16 / 30
Reflection Arrangements A 3 B 3 H 3 Remark ◮ Does not depend on the choice of a base chamber c. ◮ Paths corresponds to reduced words. 17 / 30
◮ Dual hyperplane configuration is a ( n − d ) × n matrix. ◮ Functions on A correspond to dependencies of A ∗ . ◮ When n − d is small, this makes things easy. a 1 a 2 a 3 a 4 1 � a ∗ a ∗ a ∗ a ∗ 1 0 0 1 1 2 3 4 � 1 A ∗ = · 0 1 0 1 = 0 1 1 1 − 1 1 0 0 1 1 − 1 Example a ∗ 1 + a ∗ 2 + a ∗ 3 + 3 a ∗ + + ++ f ( x , y , z ) = x + y + z 4 = 0 − a ∗ 1 + a ∗ 2 + a ∗ 3 + a ∗ − + ++ f ( x , y , z ) = − x + y + z 4 = 0 + + + − ? ? Affine Gale duals replace ( A , f ) with a picture. 18 / 30
Contraction � 1 � 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 Lifting Extension Deletion Proposition ◮ Extensions preserve dimension. ◮ Liftings preserve corank; 1 0 0 if f is generic on A then 0 1 0 there exists � f is generic 0 0 1 on � A . 19 / 30
Contraction Lifting Extension Deletion Proposition If γ is an f-monotone path of A and � A a single-element lifting of A , then any � γ with � γ/ ( n + 1 ) = γ is an � f-monotone path of � A . 20 / 30
Contraction Lifting Extension Deletion Proposition If A + is a single-element extension of A , and γ + is an f-monotone path of A + then any γ \ ( n + 1 ) is an f-monotone path of A . 21 / 30
Findings: Reflection Arrangements A | Γ( A ) | H 3 152 D 4 2316 D 5 12985968 D 6 3705762080 F 4 2144892 Proposition H 3 has exactly 4 L 2 -accessible nodes. 22 / 30
Findings: Diameter There is an ( A , f ) pair with no L 2 -accessible nodes. Example Z ( 8 , 4 ) , cyclic arrangement of 8 vectors in R 4 has Diam G 2 ( A , c ) = 30 but | L 2 | = 28 for c = ( − ) 4 (+) 4 . Theorem When n − d = 1 G 2 ( A , f ) has diameter | L 2 | and always has an L 2 -accessible node. 23 / 30
Findings: Classification of ( A , f ) in corank 1. ◮ The purple ( A , f ) pair is a minimal obstruction , all other ( A , f ) containing incoherent f -monotone paths are liftings of it. ◮ Really remarkable: Coherence depends only on the oriented matroid structure, not on the particular f . Theorem When n − d = 1 there is a unique family of all-coherent ( A , f ) pairs and all other ( A , f ) pairs have incoherent paths. 24 / 30
Findings: Classification of ( A , f ) in corank 2. Theorem When n − d = 2 there are two all-coherent families and 9 minimal obstructions. Of the 9 minimal obstructions 8 are single-element lifting of the corank 1 minimal obstruction. 25 / 30
Findings: Minimal obstructions for Cyclic Zonotopes a 1 a 2 · · · a n 1 1 · · · 1 t 1 t 2 · · · t n A ( n , d ) = , . . . . . . . . . t d − 1 t d − 1 t d − 1 · · · n 1 2 Theorem When d > 2 and f realizing c, the monotone path graph ◮ When n − d = 1 , every f-monotone path of ( A ( n , d ) , f ) is coherent when c is a reorientation of a certain hyperplane arrangement, and has incoherence f-monotone paths for all other c. ◮ When n − d ≥ 2 , ( A ( n , d ) , f ) has incoherent galleries for every f. 26 / 30
Lemma (4.17) Suppose A + = { a i , . . . , a n + 1 } is a single-element extension of A and f is a generic function on both Z ( A ) and Z ( A + ) . If γ + is a coherent f-monotone path of ( A + , f ) then γ = γ + \ ( n + 1 ) is a coherent f-monotone path of ( A , f ) . Lemma (4.22) Let A be a hyperplane arrangement and � A a single element lifting of A . Suppose γ g = ( n + 1 , 1 , 2 , . . . , n ) � � γ h = ( 1 , 2 , . . . , n , n + 1 ) are coherent � A ) , � f ) for some � f-monotone paths of ( Z ( � f. Then there is a generic functional f on Z ( A ) for which γ is a coherent f-monotone path. 27 / 30
Corank 3 Universally All-Coherent ? All-Coherent Has incoherent path ? Minimal Obstructions Lemma 4.22 Lemma 4.17 Lemma 6.6 Corank 2 Single-Element Extension Lemma 6.4 Lemma 6.2 Corank 1 Lemma 5.5 Corank 0 Billera & Sturmfels 1994 Single-Element Lifting 28 / 30
Questions? 29 / 30
Thank You. Committee Members Victor Reiner Alexander Voronov Pavlo Pylyavskyy Kevin Leder 30 / 30
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