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Topology ofGraphic

HyperplaneArrangements

KennethAscher &DonaldMathers

IntroductionDefinitionsExampleMoreDefinitionsGraphs

AlgebraO.S. AlgebraResonanceVarieties

ResultsResonanceVarietiesPolymatroidExampleFutureFuture

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Topology of Graphic Hyperplane Arrangements

Kenneth Ascher & Donald Mathers

Brown SUMS 2012

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Definitions

Setup: i = 1, · · · ,nαi : C`→ C is a non-zero linear transformationai = [ai1 · · ·ai` ]

Definition

Ker(αi ) = {x ∈ C` | αi (x) = 0} := Hiis called a linear hyperplane.

Definition

We call A = {H1, · · · ,Hn}, a finite set of hyperplanes, ahyperplane arrangement.

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Example

Example

A =

1 0 00 1 00 0 1−1 1 0−1 0 10 −1 1

H1 : x = 0H2 : y = 0H3 : z = 0H4 :−x + y = 0H5 :−x + z = 0H6 :−y + z = 0

Example

6

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Combinatorial Definitions

Definition

A subset S ⊆A is called dependent iff the set {αi | Hi ∈ S} isa linear dependent set.

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Graphic Arrangements

We will deal with graphic arrangements – arrangementsassociated with a simple graph, ΓGiven a graph, Γ with edge set E , the arrangement isformed by the following hyperplanes (in C`):

AΓ = {zi − zj = 0 | (i , j) ∈ E }

Example

Edges – HyperplanesRank of graph – Space we are in

We are interested in the complement of A ,

M = C`−n⋃

i=1

Hi (or in complex projective space. . . )

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Orlik-Solomon Algebra

We are interested in the complement of A ,

M = C`−n⋃

i=1

Hi (or in complex projective space)

For each Hi ∈A we have a corresponding basis vector, ei .Let E = ∧(e1, · · · ,e`) be the exterior algebraDefine ∂ : E p→ E p−1 by:

∂ (ei1 ∧·· ·∧ eip ) =p

∑k=1

(−1)k−1ei1 ∧·· ·∧ eik ∧·· ·∧ eip

Notation: S = (i1, · · · , ip), denote ei1 ∧·· ·∧ eip as eS

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O.S. Algebra Cont.

Let I be the ideal generated by: (∂eS | S is dependent)

Definition

The Orlik-Solomon Algebra is A(A ) = E /I

A is a homogeneous, graded algebra

Theorem

Let M be the complement as before. ThenA(A ) = E /I ∼= H∗(M)

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Resonance Varieties

Fix a ∈ A1 so that a =n

∑i=1

λiei , δa(x) = a∧ x

A(A ) is graded so we have:

0→ A0 δ0a−→ A1 δ1

a−→ A2 δ2a−→ ·· · δ `−1

a−−→ A` δ `a−→ 0

We have a co-chain complex ⇒ Hk(a,δa) = Ker(δ ka )/Im(δ k−1

a )

Definition

The dth resonance variety isRd (A ) = {a ∈ Ad | Hd (Ad ,δ d

a ) 6= 0}

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Resonance Varieties, Cont.

Important properties:

It is an algebraic variety, specifically, a union of linearsubspaces

For graphic arrangements we have the following theorem:

Theorem

For a graphic arrangement, AΓ, R1(AΓ) has a component foreach K3 and K4 in the graph.

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Graph Operations

ExampleExample

Parallel-connection vs. Parallel-indecomposable

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Resonance Varieties

We are interested in the dimension of the (first) resonancevariety and have the following theorem:

Theorem

Given a graphic arrangement, AΓ, let B be the arrangementformed by removing all hyperplanes in A which are notcontained in a K3. Call this resulting graph associated to B, H.Then, dim(R1(A )) = e(H)− c(H), where e(H) denotes thenumber of edges in H, and c(H) denotes the number ofmaximal edge-joint components.

Corollary

Given a 2-connected, parallel-indecomposable graph Γ, suchthat each edge is contained in a K3, dim(R1(A )) = e(Γ)−1.

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Resonance Varieties

Example

This is what we call aparallel connection. Inthis case, we parallelconnected two K3s.The dimension of thespan of R1(A ) is 4 .

Example

This is a W5, a wheelgraph with 5 vertices.The dimension of thespan of R1(A ) is 7 .

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Polymatroids

We’ve already established that the resonance variety is a unionof linear subspaces.

Let’s assume R1(A ) = L1∪L2 · · ·∪Lk⋃

M1∪·· ·∪Mn

Li ↔ K3Mi ↔ K4

The polymatroid is a function that assigns to eachsubspace of R1, the dimension of its span

Using our theorem on dimension of the span of the resonancevariety, we can calculate the polymatroid of a graph containingno 4-cliques.

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Polymatroid

Let Γ be a K4-free graph (a decomposable arrangement)

Each K3 contributes a local component, a linear space ofdimension 2, so we are interested in the dimension of thespan of the union of any componentsWe call the dimension of the span degenerate if it is “lessthan expected“Wheel graphs represent “minimal degenerate sets“

We can determine the polymatroid of these arrangements bylooking at subgraphs which are wheel graphs.

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Examples

Example

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Example

Example

Example

Parallel-indecomposable, irreducible, inerectibleSame chromatic polynomial and same polymatroidSame quadratic O.S. algebras

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Future

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Acknowledgements

Professor Michael FalkCaleb Holtzinger

YMCNSF

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Thanks!

Questions?

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