# Math 580AMatroids and Hyperplane Arrangements

## Fall 2019 – Spring 2020

### Course Mechanics

#### Teacher

Thomas Zaslavsky
Office: WH-216
Email: zaslav@math.binghamton.edu

Office hours by appointment.

#### Course goal

Develop theoretical understanding of advanced combinatorial linear algebra. (I adapted this from my 304 linear algebra class, and it fits.)

#### Syllabus

We will cover the following, Chronos willing:
• Arrangements of hyperplanes, mainly in real and complex space.
• Dually, arrangements of points and positive span.
• The matroid theory behind hyperplane arrangements.
• Matroid invariants for arrangements.
• Arrangements described by graphs, signed graphs, and gain graphs.
• The commutative algebra of arrangements (the Orlik-Solomon algebra) and cohomology of the complex complement.
Sources:

#### Written homework assignments

• I will assign and grade some written homework. [This turned out to be taking notes on gain graphs and their arrangements. No paper was required.]
• Staple multiple sheets (paper clips and paper folds don't work).
• Write your solutions by yourself in your own words. You may discuss the problems before writing up the solution, but you must write in your own words and formulas.

#### Exam policy

There will be no written exams. There will be periodic oral exams.

### Errata and Addenda for Stanley

That is, besides those in Stanley's E&A.
• Example 1.2: This is not the "sweep hyperplane" method. The sweep line method (in general dimension, the "sweep hyperplane" method) is an entirely different method.
• Prop. 1.1, proof: "By linear algebra, ..., then \$H \cap x = x\$ or \$H \cap x = \emptyset\$ or \$\dim(x) - \dim(H \cap x) = 1\$, ..."
• Geometry of relative vertices: proof that they are translates of each other.
• Lecture 5, page 3 [p. 63], middle: An−1 is not Bn. The former is a set of vectors. The latter is the dual hyperplane arrangement, for which better notation is An*. When A is a set of vectors, I will write A* for the dual hyperplane arrangement.

### Lecture Notes on Gain Graphs and Arrangements

Here are the fully revised (up to now) combined lecture notes in PDF. The LaTeX and the separate sections are linked from a separate page, and they are not kept up to date.

### The Orlik–Solomon Algebra

In Orlik's booklet you will find the definition of the algebra A at the beginning of Ch. 3. I will develop it for a simple matroid, which is more general than Orlik's treatment for an arrangement, but the properties and proofs are the same.
Some missing proofs are provided in my
notes on A(M), M a simple matroid.
We'll omit the algebra B, which follows A in Ch. 3. (B is isomorphic to A but more complicated.)
The connection to the complex complement is in Theorem 6.3.

#### Exercises for OS

1. We have a simple matroid M of rank l. Find the algebra A(M) when
• l = 1 (easy).
• l = 2 (cute; it's in Orlik and we do it in class).
• l = 3 (a serious exercise, begun in class).
l = 4 is far too complicated for a mere exercise.
2. ?

### Assignments

Dates are due dates. Sections §s.n mean readings. Boldface problem numbers (s.n) are to be handed in. Problem numbers (s.n) are particularly (but not exclusively) to work for yourself.

Warning: This list is not up to date. See the emails for full assignments.

• F 8/23: §§1.1-1.2.
• M 8/26: Reread §§1.1-1.2 carefully. Start §1.3.
• Late Sept.: Finish reading Lecture 2.
• W 10/2: §3.1.
• F 10/4: §3.2.

• F 1/24: Solve the characteristic polynomial of the hollow Catalan arrangement, Cn° = A[{±1}Kn].
Optional: Solve that of the extended Catalan arrangement, Cn,k = A[{0,±1,...,±k}Kn].
• (M 1/27, i.e.) W 1/29: §§5.1-5.2.
• M 2/3: §§5.4, 5.7, and 5.3.
• W 2/5: §§5.3 and parts of 5.5 that are about arrangements (starting at the bottom of page 461 in the book, page 77 in the separate file), not interval orders. You may omit the proof of Theorem 5.19.
• M 2/10: §5.6 (but I will omit this from the lectures).