Introduction To Hyperplane Arrangements

An Introduction to Hyperplane Arrangements Richard P. Stanley Contents An Introduction to Hyperplane Arrangements 1 Lecture 1. Basic definitions, the intersection poset and the characteristic polynomial Exercises 2 12 Lecture 2. Properties of the intersection poset and graphical arrangements Exercises 13 30 Lecture 3. Matroids and geometric lattices Exercises 31 39 Lecture 4. Broken circuits, modular elements, and supersolvability Exercises 41 58 Lecture 5. Finite fields Exercises 61 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 definitions, the intersection poset and the characteristic polynomial 1.1. Basic definitions The following notation is used throughout for certain sets of numbers: N P Z Q R R+ C [m] nonnegative integers positive integers integers rational numbers real numbers positive real numbers complex numbers the set {1, 2, . . . , m} when m ∈ N We also write [tk ]χ(t) for the coefficient of tk in the polynomial or power series χ(t). For instance, [t2 ](1 + t)4 = 6. A finite hyperplane arrangement A is a finite set of affine hyperplanes in some vector space V ∼ = K n , where K is a field. We will not consider infinite 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 finite 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 fields as well. To make sure that the definition of a hyperplane arrangement is clear, we def