Coherence of f -Monotone Paths onZonotopes.

Robert Edman

May 15, 2015

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An Analogy: The Secondary Polytope

Definition (Polytope)

A polytope is a convex hull of finitely many points in Rd .Combinatorially a polytope can be defined by its face lattice.

Definition (Polyhedral Subdivision)

A polyhedral subdivision is a decomposition of P intosubpolytopes. A subdivision is a triangulation when eachsubpolytope is a simplex.

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RemarkSubdivisions of P form a poset called the refinement poset of P.

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RemarkIn this example, the refinement poset is the face lattice of apolytope.

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I Some bad triangulations are not regular or are incoherent.I Coherence is a linear inequality condition.I Σ(P) is an example of a Fiber Polytope.

Theorem (GKZ)

The refinement poset of all regular subdivisions of P is the facelattice of a polytope Σ(P).

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Our Work: Monotone Paths

v1

v2v3

v4

v5

f

I Our version of triangulationsare f -monotone edge paths ofP .

I f must be generic,non-constant on each edge ofP.

I The refinement poset consistsof cellular strings.

DefinitionAn f -monotone edge path is a path from the f -minimal vertex−z to the f -maximal vertex z along the edges of P.

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Definition

I The vertices graph G2(P, f ) is formed from all elements onthe bottom level levels of the refinement poset.

I In this example every f -monotone path is coherent.

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QuestionWhen does P have incoherent f -monotone paths?

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Definition (Coherent)

An f -monotone path γ is coherent if there exists g ∈(Rd)∗

making γ the lower face of the polytopeP = Conv {(f (pi),g(pi))} ⊂ R2.

3

2

2

22

1

3

1

3

1

1 3

−z

z

f

3

2

22

2 1

3

1

3

1

1

3

−z z

f

g

RemarkThe refinement poset of coherent cellular strings is the fiberpolytope Σ(P, f ).

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1

2

4

3

f

1324

4321

1342

3142

3412

3214

1423

2341

4123

1243

3124

3421 2431

1234

4312 4213

2134

4231

1432

3241

2413

2143

2314

4132

Theorem (Billera & Sturmfels)

Every f -monotone path of a cube is coherent.

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+−−−−−−−

12

43

−−−−+−−− 421

3

A

Definition

I A zonotope is the image of then-cube in Rd under aprojection A : Cn → Rd

specified by a d × n matrix

A =

| | |a1 a2 . . . an| | |

I The zonotope

Z (A) =∑

[−ai ,+ai ] is theMinkowski of the columns ofA.

I The vertices of Z (A) are signvectors

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Proposition

I Every f -monotone path of Z (A) is of length n.I The function f is generic when f (ai) > 0 for all i .I The choice of f corresponds to the choice of a f -minimal

vertex −z.I But not all vertices are symmetric, so we will have to

consider multiple options for z.I The corank of Z is n − d.

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f

γ(1)

γ(2)

γ(4)

γ(3)

−z

z

g

f

gγ(1)fγ(1)

gγ(2)fγ(2)

gγ(4)fγ(4)

gγ(3)fγ(3)

Proposition

A f -monotone path γ is coherent if there exists a g ∈(Rd)∗ so

that:gγ(1)

fγ(1)<

gγ(2)

fγ(2)< . . . <

gγ(n)

fγ(n)

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Corank 1

Z (4,3) =

a1 a2 a3 a4

1 1 1 11 2 3 41 4 9 16

−+ ++ + + ++

Remark

I Every f -monotone path is coherent for −+ ++.I + + ++ has an incoherent f -monotone path for every f .

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Corank 2 (cyclic)

Z (5,3) =

a1 a2 a3 a4 a5

1 1 1 1 11 2 3 4 51 4 9 16 25

−+ + + + −−+ + + + + + + +

Remark

I Has incoherent f -monotone path for every f .I + + + + + is an important geometric counterexample.

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Definition (Pointed hyperplane arrangement)

The normal fan of the zonotope, is a hyperplane arrangement,A =

{a⊥1 , . . . ,a

⊥n}

. The choice of a chamber c of Acorresponds to the choice of f .

− − − − −

a⊥1

a⊥2

a⊥3

a⊥4

a⊥5

X2,3

I Easy to draw understereographic projection

I k -faces of Z ⇐⇒ d − kintersections ofhyperplanes.

I L2(A) are the codimension2 intersections ofhyperplanes.

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Reflection Arrangements

A3 B3 H3

Remark

I Does not depend on the choice of a base chamber c.I Paths corresponds to reduced words.

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I Dual hyperplane configuration is a (n − d)× n matrix.I Functions on A correspond to dependencies of A∗.I When n − d is small, this makes things easy.

a1 a2 a3 a4

1 0 0 10 1 0 10 0 1 1

·

111−1

= 0 A∗ =(a∗1 a∗2 a∗3 a∗4

1 1 1 −1)

Example+ + ++ f (x , y , z) = x + y + z a∗1 + a∗2 + a∗3 + 3a∗4 = 0−+ ++ f (x , y , z) = −x + y + z −a∗1 + a∗2 + a∗3 + a∗4 = 0+ + +− ? ?

Affine Gale duals replace (A, f ) with a picture.

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Lifting

Contraction

Del

etio

n

Ext

ensi

on

1 0 0 10 1 0 10 0 1 1

(1 0 10 1 1

)

1 0 00 1 00 0 1

Proposition

I Extensions preservedimension.

I Liftings preserve corank;if f is generic on A thenthere exists f is genericon A.

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Lifting

Contraction

Del

etio

n

Ext

ensi

on

Proposition

If γ is an f -monotone path ofA and A a single-elementlifting of A, then any γ withγ/(n + 1) = γ is anf -monotone path of A.

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Lifting

Contraction

Del

etio

n

Ext

ensi

on

Proposition

If A+ is a single-elementextension of A, and γ+ is anf -monotone path of A+ thenany γ\(n + 1) is anf -monotone path of A.

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Findings: Reflection Arrangements

A |Γ(A)|H3 152D4 2316D5 12985968D6 3705762080F4 2144892

Proposition

H3 has exactly 4 L2-accessible nodes.

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Findings: Diameter

There is an (A, f ) pair withno L2-accessible nodes.

Example

Z (8,4), cyclic arrangementof 8 vectors in R4 hasDiam G2(A, c) = 30 but|L2| = 28 for c = (−)4(+)4.

TheoremWhen n − d = 1 G2(A, f ) has diameter |L2| and always has anL2-accessible node.

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Findings: Classification of (A, f ) in corank 1.

I The purple (A, f ) pair is aminimal obstruction, all other(A, f ) containing incoherentf -monotone paths are liftingsof it.

I Really remarkable:Coherence depends only onthe oriented matroid structure,not on the particular f .

TheoremWhen n − d = 1 there is a unique family of all-coherent (A, f )pairs and all other (A, f ) pairs have incoherent paths.

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Findings: Classification of (A, f ) in corank 2.

TheoremWhen n − d = 2 there are two all-coherent families and 9minimal obstructions. Of the 9 minimal obstructions 8 aresingle-element lifting of the corank 1 minimal obstruction.

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Findings: Minimal obstructions for Cyclic Zonotopes

A(n,d) =

a1 a2 · · · an

1 1 · · · 1t1 t2 · · · tn...

......

td−11 td−1

2 · · · td−1n

,

TheoremWhen d > 2 and f realizing c, the monotone path graph

I When n − d = 1, every f -monotone path of (A(n,d), f ) iscoherent when c is a reorientation of a certain hyperplanearrangement, and has incoherence f -monotone paths forall other c.

I When n − d ≥ 2, (A(n,d), f ) has incoherent galleries forevery f .

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Lemma (4.17)

Suppose A+ = {ai , . . . ,an+1} is a single-element extension ofA and f is a generic function on both Z (A) and Z (A+). If γ+ isa coherent f -monotone path of (A+, f ) then γ = γ+\(n + 1) is acoherent f -monotone path of (A, f ).

Lemma (4.22)

Let A be a hyperplane arrangement and A a single elementlifting of A. Suppose

γg = (n + 1,1,2, . . . ,n)

γh = (1,2, . . . ,n,n + 1)

are coherent f -monotone paths of (Z (A), f ) for some f . Thenthere is a generic functional f on Z (A) for which γ is a coherentf -monotone path.

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Lemma 6.2

MinimalObstructions

Universally All-Coherent

Has incoherent path

Cora

nk

2

Cora

nk

3

Cora

nk

1?

Lemma 6.4

Lemma 5.5

All-Coherent

Single-Element Lifting

Sin

gle

-Ele

ment

Exte

nsi

on

Corank 0 Billera & Sturmfels 1994

Lemma 6.6

?

Lemma 4.22Lemma 4.17

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Questions?

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Thank You.

Committee MembersVictor Reiner Alexander VoronovPavlo Pylyavskyy Kevin Leder

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