18.315 | Fall 2004 | Graduate

Combinatorial Theory: Hyperplane Arrangements

Pages

Difficulty Rating of Problems

I have included a difficulty level for each problem as follows:

  1. Easy: most students should be able to solve it
  2. Moderately difficult: many students should be able to solve it
  3. Difficult: a few students should be able to solve it
  4. Horrendous: no students should be able to solve it (without already knowing how)
  5. Unsolved

Further gradations are indicated by + and -. Thus a [3-] problem is about the most difficult problem that makes a reasonable homework exercise, and a [5-] problem is an unsolved problem that has received little attention and may not be too difficult.

“Reasonable” collaboration is permitted on problems unless stated otherwise. You should not simply copy another person’s solution or obtain a solution from an outside source. If you have collaborated on a problem, please state so (with the name of the collaborator(s)) on your problem set.

The problem sets are due during the lecture noted in the table below.

ASSIGNMENTS INSTRUCTIONS DUE DATES
Problem Set 1 (PDF) Hand in two from 1.1-1.7. Ideally you should work on all seven problems, but you should hand in only your two “best” problems, i.e., the ones whose solutions you feel are the most difficult and/or interesting Lecture 7
Problem Set 2 (PDF) Hand in two from 2.1-2.10 Lecture 11
Problem Set 3 (PDF) Hand in two from 3.1-3.7, 4.1, 4.3, 4.10-4.11, 4.20 Lecture 16
Problem Set 4 (PDF) Hand in one from 4.4-4.9, 4.12-4.19, and one from 5.1, 5.4, 5.6-5.9 Lecture 22
Problem Set 5 (PDF) Hand in two from 5.10-5.18, 5.24-5.29 Lecture 27
Problem Set 6 (PDF) Hand in 6.1 and one further problem from 6.3-6.8 Lecture 32
Problem Set 7 Make up two problems of your own and hand them in (with solutions). Problems may have multiple parts. Some of the parts can be unsolved (or, at least, you were unable to solve them), but each of the two problems should have at least one part with a (nontrivial) solution included. The problems can be on any material covered during the course Lecture 37

# OF LECTURES TOPICS
3 Basic Definitions, the Intersection Poset and the Characteristic Polynomial
4.5 Properties of the Intersection Poset and Graphical Arrangements
3.5 Matroids and Geometric Lattices
4.5 Broken Circuits, Modular Elements, and Supersolvability
10 Finite Fields
3.5 Random Walks on Regions
5 Separating Hyperplanes
3 Tutte Polynomial

These lecture notes on hyperplane arrangements are based on a lecture series at the Park City Mathematics Institute, July 12-19, 2004. They provide an introduction to hyperplane arrangements, focusing on connections with combinatorics, at the beginning graduate student level. Background material on posets and matroids is included, as well as numerous exercises. After going through these notes a student should be ready to study the deeper algebraic and topological aspects of the theory of hyperplane arrangements. Perhaps someday these notes will be expanded into a textbook on arrangements.

Chapter 1: Basic Definitions, the Intersection Poset and the Characteristic Polynomial (PDF)

1.1. Basic Definitions
1.2. The Intersection Poset
1.3. The Characteristic Polynomial

Chapter 2: Properties of the Intersection Poset and Graphical Arrangements (PDF)

2.1. Properties of the Intersection Poset
2.2. The Number of Regions
2.3. Graphical Arrangements

Chapter 3: Matroids and Geometric Lattices (PDF)

3.1. Matroids
3.2. The Lattice of Flats and Geometric Lattices

Chapter 4: Broken Circuits, Modular Elements, and Supersolvability (PDF)

4.1. Broken Circuits
4.2. Modular Elements
4.3. Supersolvable Lattices

Chapter 5: Finite Fields (PDF)

5.1. The Finite Field Method
5.2. The Shi Arrangement
5.3. Exponential Sequences of Arrangements
5.4. The Catalan Arrangement
5.5. Interval Orders
5.6. Intervals with Generic Lengths
5.7. Other Examples

Chapter 6: Separating Hyperplanes (Preliminary Version) (PDF)

6.1. The Distance Enumerator
6.2. Parking Functions and Tree Inversions
6.3. The Distance Enumerator of the Shi Arrangement
6.4. The Distance Enumerator of a Supersolvable Arrangement
6.5. The Varchenko Matrix

Below is a bibliography for Hyperplane Arrangements.

Athanasiadis, C. A. “Algebraic Combinatorics of Graph Spectra, Subspace Arrangements and Tutte Polynomials.” Ph.D. Thesis, MIT, 1996.

———. “Characteristic Polynomials of Subspace Arrangements and Finite Fields.” Advances in Math. 122 (1996): 193-233.

Barcelo, H., and E. Ihrig. “Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements.” J. Algebraic Combinatorics 9 (1999): 5-24.

Björner, A., and F. Brenti. Combinatorics of Coxeter Groups. (To appear)

Björner, A., M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented Matroids. 2nd ed_. Encyclopedia of Mathematics and Its Applications_ 46_._ Cambridge: Cambridge University Press, 1999.

Blass, A., and B. Sagan. “Characteristic and Ehrhart Polynomials.” J. Algebraic Combinatorics 7 (1998): 115-126.

Bourbaki, N. “Groupes et Algèbres de Lie.” Éléments de Mathématique, Fasc. 34. Paris: Hermann, 1968.

Chandon, J. L., J. Lemaire, and J. Pouget. “Dénombrement des Quasi-ordres sur un Ensemble fini.” Math. Inform. Sci. Humaines 62 (1978): 61-80, 83.

Crapo, H., and G.-C. Rota. On the Foundations of Combinatorial Theory: Combinatorial Geometries. Preliminary edition. Cambridge, MA: MIT Press, 1970.

Fishburn, P. C. Interval Orders and Interval Graphs. New York: Wiley-Interscience, 1985.

Greene, C. “On the Möbius Algebra of a Partially Ordered Set.” Advances in Math. 10 (1973): 177-187.

Humphreys, J. E. Reflection Groups and Coxeter Groups. Cambridge: Cambridge University Press, 1990.

Orlik, P., and H. Terao. Arrangements of Hyperplanes. Berlin: Springer-Verlag, 1992.

Postnikov, A., and R. Stanley. “Deformations of Coxeter Hyperplane Arrangements.” J. Combinatorial Theory (A) 91 (2000): 544-597.

Shi, J.-Y. “The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups.” Lecture Notes in Mathematics. Vol. 1179. Berlin: Springer-Verlag, 1986.

Stanley, R. “Modular Elements of Geometric Lattices.” Algebra Universalis 1 (1971): 214-217.

———. “Supersolvable Lattices.” Algebra Universalis 2 (1972): 197-217.

———. Enumerative Combinatorics. Vol. 1. Pacific Grove, CA: Wadsworth and Brooks/Cole, 1986. Second printing, Cambridge: Cambridge University Press, 1996.

———. Enumerative Combinatorics. Vol. 2. Cambridge: Cambridge University Press, 1999.

———. “Hyperplane Arrangements, Interval Orders, and Trees.” Proc. Nat. Acad. Sci. 93 (1996): 2620-2625.

Terao, H. “Free Arrangements of Hyperplanes and Unitary Reflection Groups.” Proc. Japan Acac., Ser. A 56 (1980): 389-392.

———. “Generalized Exponents of a Free Arrangement of Hyperplanes and Shepherd[sic]-Todd-Brieskorn Formula.” Invent. Math. 63 (1981): 159-179.

Trotter, W. T. Combinatorics and Partially Ordered Sets. Baltimore/London: The Johns Hopkins Univ. Press, 1992.

Wine, R. L., and J. E. Freund. “On the Enumeration of Decision Patterns Involving n Means.” Ann. Math. Stat. 28: 256-259.

Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Course Information

About half the course will be based on notes from a previous lecture series. Some of the topics covered in the course involve connections with topology, representation theory, and commutative algebra. Those with insufficient backgrounds in these topics should still be able to follow most of the course.

Textbook

There is no required text, although I have included an extensive bibliography.

The main text on hyperplane arrangements is: Orlik, P., and H. Terao. Arrangements of Hyperplanes. New York: Springer-Verlag, 1992.

This book has more emphasis on algebra and topology and less on combinatorics than will be presented in the course. There is no real reason to purchase this book for the course, but those who want to look deeper into the subject can do so.

Grading

Grading will be based on problem sets due approximately once every two weeks.

Course Info

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As Taught In
Fall 2004
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Learning Resource Types
Lecture Notes
Problem Sets