Arrangements

Richard P. Stanley

Contents

1

An Introduction to Hyperplane Arrangements

Lecture 1. Basic de¬nitions, the intersection poset and the characteristic

polynomial 2

Exercises 12

Lecture 2. Properties of the intersection poset and graphical arrangements 13

Exercises 30

Lecture 3. Matroids and geometric lattices 31

Exercises 39

Lecture 4. Broken circuits, modular elements, and supersolvability 41

Exercises 58

Lecture 5. Finite ¬elds 61

Exercises 81

Bibliography 89

3

IAS/Park City Mathematics Series

Volume 00, 0000

An Introduction to Hyperplane

Arrangements

Richard P. Stanley

1

2 R. STANLEY, HYPERPLANE ARRANGEMENTS

LECTURE 1

Basic de¬nitions, the intersection poset and the

characteristic polynomial

1.1. Basic de¬nitions

The following notation is used throughout for certain sets of numbers:

nonnegative integers

N

positive integers

P

integers

Z

rational numbers

Q

real numbers

R

positive real numbers

R+

complex numbers

C

the set {1, 2, . . . , m} when m ∈ N

[m]

We also write [tk ]χ(t) for the coe¬cient of tk in the polynomial or power series χ(t).

For instance, [t2 ](1 + t)4 = 6.

A ¬nite hyperplane arrangement A is a ¬nite set of a¬ne hyperplanes in some

vector space V ∼ K n , where K is a ¬eld. We will not consider in¬nite hyperplane

=

arrangements or arrangements of general subspaces or other objects (though they

have many interesting properties), so we will simply use the term arrangement for

a ¬nite hyperplane arrangement. Most often we will take K = R, but as we will see

even if we™re only interested in this case it is useful to consider other ¬elds as well.

To make sure that the de¬nition of a hyperplane arrangement is clear, we de¬ne a

linear hyperplane to be an (n ’ 1)-dimensional subspace H of V , i.e.,

H = {v ∈ V : ± · v = 0},

where ± is a ¬xed nonzero vector in V and ± · v is the usual dot product:

(±1 , . . . , ±n ) · (v1 , . . . , vn ) = ±i v i .

An a¬ne hyperplane is a translate J of a linear hyperplane, i.e.,

J = {v ∈ V : ± · v = a},

where ± is a ¬xed nonzero vector in V and a ∈ K.

If the equations of the hyperplanes of A are given by L1 (x) = a1 , . . . , Lm (x) =

am , where x = (x1 , . . . , xn ) and each Li (x) is a homogeneous linear form, then we

call the polynomial

QA (x) = (L1 (x) ’ a1 ) · · · (Lm (x) ’ am )

the de¬ning polynomial of A. It is often convenient to specify an arrangement

by its de¬ning polynomial. For instance, the arrangement A consisting of the n

coordinate hyperplanes has QA (x) = x1 x2 · · · xn .

Let A be an arrangement in the vector space V . The dimension dim(A) of

A is de¬ned to be dim(V ) (= n), while the rank rank(A) of A is the dimension

of the space spanned by the normals to the hyperplanes in A. We say that A is

essential if rank(A) = dim(A). Suppose that rank(A) = r, and take V = K n . Let

LECTURE 1. BASIC DEFINITIONS 3

Y be a complementary space in K n to the subspace X spanned by the normals to

hyperplanes in A. De¬ne

W = {v ∈ V : v · y = 0 ∀y ∈ Y }.

If char(K) = 0 then we can simply take W = X. By elementary linear algebra we

have

codimW (H © W ) = 1

(1)

for all H ∈ A. In other words, H © W is a hyperplane of W , so the set AW :=

{H ©W : H ∈ A} is an essential arrangement in W . Moreover, the arrangements A

and AW are “essentially the same,” meaning in particular that they have the same

intersection poset (as de¬ned in De¬nition 1.1). Let us call AW the essentialization

of A, denoted ess(A). When K = R and we take W = X, then the arrangement A

is obtained from AW by “stretching” the hyperplane H © W ∈ AW orthogonally to

W . Thus if W ⊥ denotes the orthogonal complement to W in V , then H ∈ AW if

and only if H • W ⊥ ∈ A. Note that in characteristic p this type of reasoning fails

since the orthogonal complement of a subspace W can intersect W in a subspace

of dimension greater than 0.

Example 1.1. Let A consist of the lines x = a1 , . . . , x = ak in K 2 (with coordinates

x and y). Then we can take W to be the x-axis, and ess(A) consists of the points

x = a1 , . . . , x = ak in K.

Now let K = R. A region of an arrangement A is a connected component of

the complement X of the hyperplanes:

X = Rn ’ H.

H∈A

Let R(A) denote the set of regions of A, and let

r(A) = #R(A),

the number of regions. For instance, the arrangement A shown below has r(A) = 14.

It is a simple exercise to show that every region R ∈ R(A) is open and convex

(continuing to assume K = R), and hence homeomorphic to the interior of an n-

dimensional ball Bn (Exercise 1). Note that if W is the subspace of V spanned by

the normals to the hyperplanes in A, then R ∈ R(A) if and only if R ©W ∈ R(AW ).

We say that a region R ∈ R(A) is relatively bounded if R © W is bounded. If A

is essential, then relatively bounded is the same as bounded. We write b(A) for

4 R. STANLEY, HYPERPLANE ARRANGEMENTS

the number of relatively bounded regions of A. For instance, in Example 1.1 take

K = R and a1 < a2 < · · · < ak . Then the relatively bounded regions are the

regions ai < x < ai+1 , 1 ¤ i ¤ k ’ 1. In ess(A) they become the (bounded) open

intervals (ai , ai+1 ). There are also two regions of A that are not relatively bounded,

viz., x < a1 and x > ak .

A (closed) half-space is a set {x ∈ Rn : x · ± ≥ c} for some ± ∈ Rn , c ∈ R. If

H is a hyperplane in Rn , then the complement Rn ’ H has two (open) components

¯

whose closures are half-spaces. It follows that the closure R of a region R of A is

a ¬nite intersection of half-spaces, i.e., a (convex) polyhedron (of dimension n). A

¯

bounded polyhedron is called a (convex) polytope. Thus if R (or R) is bounded,

¯

then R is a polytope (of dimension n).