![Topology of
Graphic
Hyperplane
Arrangements
Kenneth
Ascher &
Donald
Mathers
Introduction
Definitions
Example
More
Definitions
Graphs
Algebra
O.S. Algebra
Resonance
Varieties
Results
Resonance
Varieties
Polymatroid
Example
Future
Future
End
Definitions
Setup: i = 1,··· ,n
αi : C → C is a non-zero linear transformation
ai = [ai1 ···ai ]
Definition
Ker(αi ) = {x ∈ C | αi (x) = 0} := Hi
is called a linear hyperplane.
Definition
We call A = {H1,··· ,Hn}, a finite set of hyperplanes, a
hyperplane arrangement.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- 1. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Brown SUMS 2012
- 2. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Definitions Setup: i = 1,··· ,n αi : C → C is a non-zero linear transformation ai = [ai1 ···ai ] Definition Ker(αi ) = {x ∈ C | αi (x) = 0} := Hi is called a linear hyperplane. Definition We call A = {H1,··· ,Hn}, a finite set of hyperplanes, a hyperplane arrangement.
- 3. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Example Example A = 1 0 0 0 1 0 0 0 1 −1 1 0 −1 0 1 0 −1 1 H1 : x = 0 H2 : y = 0 H3 : z = 0 H4 : −x +y = 0 H5 : −x +z = 0 H6 : −y +z = 0 Example 6
- 4. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Combinatorial Definitions Definition A subset S ⊆ A is called dependent iff the set {αi | Hi ∈ S} is a linear dependent set.
- 5. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Graphic Arrangements We will deal with graphic arrangements – arrangements associated with a simple graph, Γ Given a graph, Γ with edge set E, the arrangement is formed by the following hyperplanes (in C ): AΓ = {zi −zj = 0 | (i,j) ∈ E } Example
- 6. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future 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 algebra Define ∂ : E p → E p−1 by: ∂(ei1 ∧···∧eip ) = p ∑ k=1 (−1)k−1 ei1 ∧···∧eik ∧···∧eip Notation: S = (i1,··· ,ip), denote ei1 ∧···∧eip as eS
- 7. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future 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. Then A(A ) = E /I ∼= H∗(M)
- 8. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Resonance Varieties Fix a ∈ A1 so that a = n ∑ i=1 λi ei , δa(x) = a ∧x A(A ) is graded so we have: 0 → A0 δ0 a −→ A1 δ1 a −→ A2 δ2 a −→ ··· δ −1 a −−→ A δa −→ 0 We have a co-chain complex ⇒ Hk(a,δa) = Ker(δk a )/Im(δk−1 a ) Definition The dth resonance variety is Rd (A ) = {a ∈ Ad | Hd (Ad ,δd a ) = 0}
- 9. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Resonance Varieties, Cont. Important properties: It is an algebraic variety, specifically, a union of linear subspaces For graphic arrangements we have the following theorem: Theorem For a graphic arrangement, AΓ, R1(AΓ) has a component for each K3 and K4 in the graph.
- 10. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Graph Operations Example Example Parallel-connection vs. Parallel-indecomposable
- 11. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Resonance Varieties We are interested in the dimension of the (first) resonance variety and have the following theorem: Theorem Given a graphic arrangement, AΓ, let B be the arrangement formed by removing all hyperplanes in A which are not contained in a K3. Call this resulting graph associated to B, H. Then, dim(R1(A )) = e(H)−c(H), where e(H) denotes the number of edges in H, and c(H) denotes the number of maximal edge-joint components. Corollary Given a 2-connected, parallel-indecomposable graph Γ, such that each edge is contained in a K3, dim(R1(A )) = e(Γ)−1.
- 12. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Resonance Varieties Example This is what we call a parallel connection. In this case, we parallel connected two K3s. The dimension of the span of R1(A ) is 4 . Example This is a W5, a wheel graph with 5 vertices. The dimension of the span of R1(A ) is 7 .
- 13. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Polymatroids We’ve already established that the resonance variety is a union of linear subspaces. Let’s assume R1(A ) = L1 ∪L2 ···∪Lk M1 ∪···∪Mn Li ↔ K3 Mi ↔ K4 The polymatroid is a function that assigns to each subspace of R1, the dimension of its span Using our theorem on dimension of the span of the resonance variety, we can calculate the polymatroid of a graph containing no 4-cliques.
- 14. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Polymatroid Let Γ be a K4-free graph (a decomposable arrangement) Each K3 contributes a local component, a linear space of dimension 2, so we are interested in the dimension of the span of the union of any components We call the dimension of the span degenerate if it is “less than expected“ Wheel graphs represent “minimal degenerate sets“ We can determine the polymatroid of these arrangements by looking at subgraphs which are wheel graphs.
- 15. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Examples Example
- 16. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Examples Example
- 17. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Examples Example
- 18. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Examples Example
- 19. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Example Example Example Parallel-indecomposable, irreducible, inerectible Same chromatic polynomial and same polymatroid Same quadratic O.S. algebras
- 20. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Future
- 21. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Future
- 22. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Acknowledgements Professor Michael Falk Caleb Holtzinger YMC NSF
- 23. Topology of Graphic Hyperplane Arrangements Kenneth Ascher & Donald Mathers Introduction Definitions Example More Definitions Graphs Algebra O.S. Algebra Resonance Varieties Results Resonance Varieties Polymatroid Example Future Future End Thanks! Questions?