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Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Topology of Graphic Hyperplane Arrangements
Kenneth Ascher & Donald Mathers
Brown SUMS 2012
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Combinatorial Definitions
Definition
A subset S ⊆A is called dependent iff the set {αi | Hi ∈ S} isa linear dependent set.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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. . . )
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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)
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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}
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Graph Operations
ExampleExample
Parallel-connection vs. Parallel-indecomposable
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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 .
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
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.
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Examples
Example
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Examples
Example
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Examples
Example
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Examples
Example
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Example
Example
Example
Parallel-indecomposable, irreducible, inerectibleSame chromatic polynomial and same polymatroidSame quadratic O.S. algebras
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Future
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Future
Topology ofGraphic
HyperplaneArrangements
KennethAscher &DonaldMathers
IntroductionDefinitionsExampleMoreDefinitionsGraphs
AlgebraO.S. AlgebraResonanceVarieties
ResultsResonanceVarietiesPolymatroidExampleFutureFuture
End
Acknowledgements
Professor Michael FalkCaleb Holtzinger
YMCNSF
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