Basic Definitions, the Intersection Poset and the Characteristic Polynomial

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 ]ψ(t) for the coefficient of tk in the polynomial or power series ψ(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 define a linear hyperplane to be an (n − 1)-dimensional subspace H of V , i.e., H = {v ≤ V : κ · v = 0}, where κ is a fixed nonzero vector in V and κ · v is the usual dot product: � (κ1 , . . . , κn ) · (v1 , . . . , vn ) = κi v i . An affine hyperplane is a translate J of a linear hyperplane, i.e., J = {v ≤ V : κ · v = a}, where κ is a fixed 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 defining polynomial of A. It is often convenient to specify an arrangement by its defining 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 defined 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 Y be a complementary space in K n to the subspace X spanned by the normals to hyperplanes in A. Define 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 (1) codimW (H ⊕ W ) = 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 defined in Definition 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: � H. X = Rn − 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 ndimensional 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 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 · κ ⊂ c} for some κ ≤ Rn , c ≤ R. If H is a hyperplane in Rn , then the complement Rn − H has two (open) components ¯ of a region R of A is whose closures are half-spaces. It follows that the closure R a finite intersection of half-spaces, i.e., a (convex) polyhedron (of dimension n). A ¯ is bounded, bounded polyhedron is called a (convex) polytope. Thus if R (or R) ¯ then R is a polytope (of dimension n). An arrangement A is in general position if {H1 , . . . , Hp } ∗ A, p → n ⊆ dim(H1 ⊕ · · · ⊕ Hp ) = n − p {H1 , . . . , Hp } ∗ A, p > n ⊆ H1 ⊕ · · · ⊕ Hp = �. For instance, if n = 2 then a set of lines is in general position if no two are parallel and no three meet at a point. Let us consider some interesting examples of arrangements that will anticipate some later material. Example 1.2. Let Am consist of m lines in general position in R2 . We can compute r(Am ) using the sweep hyperplane method. Add a L line to Ak (with AK ∅ {L} in general position). When we travel along L from one end (at infinity) to the other, every time we intersect a line in Ak we create a new region, and we create one new region at the end. Before we add any lines we have one region (all of R2 ). Hence r(Am ) = #intersections + #lines + 1 � � m = + m + 1. 2 Example 1.3. The braid arrangement Bn in K n consists of the hyperplanes ⎜ n� Bn : xi − xj = 0, 1 → i < j → n. Thus Bn has 2 hyperplanes. To count the number of regions when K = R, note that specifying which side of the hyperplane xi − xj = 0 a point (a1 , . . . , an ) lies on is equivalent to specifying whether ai < aj or ai > aj . Hence the number of regions is the number of ways that we can specify whether ai < aj or ai > aj for 1 → i < j → n. Such a specification is given by imposing a linear order on the ai ’s. In other words, for each permutation w ≤ Sn (the symmetric group of all permutations of 1, 2, . . . , n), there corresponds a region Rw of Bn given by Rw = {(a1 , . . . , an ) ≤ Rn : aw(1) > aw(2) > · · · > aw(n) }. Hence r(Bn ) = n!. Rarely is it so easy to compute the number of regions! Note that the braid arrangement Bn is not essential; indeed, rank(Bn ) = n − 1. ⇔ 2 the space W ∗ K n of equation (1) can be taken to be When char(K) = W = {(a1 , . . . , an ) ≤ K n : a1 + · · · + an = 0}. The braid arrangement has a number of “deformations” of considerable interest. We will just define some of them now and discuss them further later. All these arrangements lie in K n , and in all of them we take 1 → i < j → n. The reader who likes a challenge can try to compute their number of regions when K = R. (Some are much easier than others.) • generic braid arrangement : xi − xj = aij , where the aij ’s are “generic” (e.g., linearly independent over the prime field, so K has to be “sufficiently large”). The precise definition of “generic” will be given later. (The prime field of K is its smallest subfield, isomorphic to either Q or Z/pZ for some prime p.) • semigeneric braid arrangement : xi −xj = ai , where the ai ’s are “generic.” • Shi arrangement : xi − xj = 0, 1 (so n(n − 1) hyperplanes in all). • Linial arrangement : xi − xj = 1. • Catalan arrangement : xi − xj = −1, 0, 1. • semiorder arrangement : xi − xj = −1, 1. • threshold arrangement : xi + xj = 0 (not really a deformation of the braid arrangement, but closely related).  ⇔ �. Equivalently, A is a translate An arrangement A is central if H⊆A H = of a linear arrangement (an arrangement of linear hyperplanes, i.e., hyperplanes passing through theorigin). Many other writers call an arrangement central, rather  than linear, if 0 ≤ H⊆A H. If A is central with X = H⊆A H, then rank(A) = codim(X). If A is central, then note also that b(A) = 0 [why?]. There are two useful arrangements closely related to a given arrangement A. If A is a linear arrangement in K n , then projectivize A by choosing some H ≤ A n−1 . Thus if we regard to be the hyperplane at infinity in projective space PK n−1 = {(x1 , . . . , xn ) : xi ≤ K, not all xi = 0}/ ∪, PK where u ∪ v if u = κv for some 0 ⇔= κ ≤ K, then n−2 H = ({(x1 , . . . , xn−1 , 0) : xi ≤ K, not all xi = 0}/ ∪) ∪ . = PK The remaining hyperplanes in A then correspond to “finite” (i.e., not at infinity) n−1 . This gives an arrangement proj(A) of hyperplanes projective hyperplanes in PK n−1 in PK . When K = R, the two regions R and −R of A become identified in proj(A). Hence r(proj(A)) = 1 2 2 r(A). When n = 3, we can draw PR as a disk with antipodal boundary points identified. The circumference of the disk represents the hyperplane at infinity. This provides a good way to visualize three-dimensional real linear arrangements. For instance, if A consists of the three coordinate hyperplanes x1 = 0, x2 = 0, and x3 = 0, then a projective drawing is given by 2 1 3 The line labelled i is the projectivization of the hyperplane xi = 0. The hyperplane at infinity is x3 = 0. There are four regions, so r(A) = 8. To draw the incidences among all eight regions of A, simply “reflect” the interior of the disk to the exterior: 12 24 34 14 23 13 Figure 1. A projectivization of the braid arrangement B4 2 2 1 3 1 Regarding this diagram as a planar graph, the dual graph is the 3-cube (i.e., the vertices and edges of a three-dimensional cube) [why?]. For a more complicated example of projectivization, Figure 1 shows proj(B 4 ) (where we regard B4 as a three-dimensional arrangement contained in the hyper­ plane x1 + x2 + x3 + x4 = 0 of R4 ), with the hyperplane xi = xj labelled ij, and with x1 = x4 as the hyperplane at infinity. We now define an operation which is “inverse” to projectivization. Let A be an (affine) arrangement in K n , given by the equations L1 (x) = a1 , . . . , Lm (x) = am . Introduce a new coordinate y, and define a central arrangement cA (the cone over A) in K n × K = K n+1 by the equations L1 (x) = a1 y, . . . , Lm (x) = am y, y = 0. For instance, let A be the arrangement in R1 given by x = −1, x = 2, and x = 3. The following figure should explain why cA is called a cone. −1 2 3 It is easy to see that when K = R, we have r(cA) = 2r(A). In general, cA has the “same combinatorics as A, times 2.” See Exercise 1. 1.2. The intersection poset Recall that a poset (short for partially ordered set) is a set P and a relation → satisfying the following axioms (for all x, y, z ≤ P ): (P1) (reflexivity) x → x (P2) (antisymmetry) If x → y and y → x, then x = y. (P3) (transitivity) If x → y and y → z, then x → z. Obvious notation such as x < y for x → y and x = ⇔ y, and y ⊂ x for x → y will be used throughout. If x → y in P , then the (closed) interval [x, y] is defined by [x, y] = {z ≤ P : x → z → y}. Note that the empty set � is not a closed interval. For basic information on posets not covered here, see [18]. Definition 1.1. Let A be an arrangement in V , and let L(A) be the set of all nonempty intersections of hyperplanes in A, including V itself as the intersection over the empty set. Define x → y in L(A) if x ∀ y (as subsets of V ). In other words, L(A) is partially ordered by reverse inclusion. We call L(A) the intersection poset of A. Note. The primary reason for ordering intersections by reverse inclusion rather than ordinary inclusion is Proposition 3.8. We don’t want to alter the well-established definition of a geometric lattice or to refer constantly to “dual geometric lattices.” The element V ≤ L(A) satisfies x ⊂ V for all x ≤ L(A). In general, if P is a poset then we denote by ˆ 0 an element (necessarily unique) such that x ⊂ ˆ 0 for all Figure 2. Examples of intersection posets x ≤ P . We say that y covers x in a poset P , denoted x � y, if x < y and no z ≤ P satisfies x < z < y. Every finite poset is determined by its cover relations. The (Hasse) diagram of a finite poset is obtained by drawing the elements of P as dots, with x drawn lower than y if x < y, and with an edge between x and y if x � y. Figure 2 illustrates four arrangements A in R2 , with (the diagram of) L(A) drawn below A. A chain of length k in a poset P is a set x0 < x1 < · · · < xk of elements of P . The chain is saturated if x0 � x1 � · · · � xk . We say that P is graded of rank n if every maximal chain of P has length n. In this case P has a rank function rk : P ∃ N defined by: • rk(x) = 0 if x is a minimal element of P . • rk(y) = rk(x) + 1 if x � y in P . If x < y in a graded poset P then we write rk(x, y) = rk(y) − rk(x), the length of the interval [x, y]. Note that we use the notation rank(A) for the rank of an arrangement A but rk for the rank function of a graded poset. Proposition 1.1. Let A be an arrangement in a vector space V ∪ = K n . Then the intersection poset L(A) is graded of rank equal to rank(A). The rank function of L(A) is given by rk(x) = codim(x) = n − dim(x), where dim(x) is the dimension of x as an affine subspace of V . ˆ = V , it suffices to show that Proof. Since L(A) has a unique minimal element 0 (a) if x�y in L(A) then dim(x)−dim(y) = 1, and (b) all maximal elements of L(A) have dimension n − rank(A). By linear algebra, if H is a hyperplane and x an affine subspace, then H ⊕ x = x or dim(x) − dim(H ⊕ x) = 1, so (a) follows. Now suppose that x has the largest codimension of any element of L(A), say codim(x) = d. Thus x is an intersection of d linearly independent hyperplanes (i.e., their normals are linearly independent) H1 , . . . , Hd in A. Let y ≤ L(A) with e = codim(y) < d. Thus y is an intersection of e hyperplanes, so some Hi (1 → i → d) is linearly independent from them. Then y ⊕ Hi ⇔= � and codim(y ⊕ Hi ) > codim(y). Hence y is not a maximal element of L(A), proving (b). � −2 1 2 −1 −1 1 −1 1 −1 1 Figure 3. An intersection poset and Möbius function values 1.3. The characteristic polynomial A poset P is locally finite if every interval [x, y] is finite. Let Int(P ) denote the set of all closed intervals of P . For a function f : Int(P ) ∃ Z, write f (x, y) for f ([x, y]). We now come to a fundamental invariant of locally finite posets. Definition 1.2. Let P be a locally finite poset. Define a function µ = µP : Int(P ) ∃ Z, called the Möbius function of P , by the conditions: µ(x, x) (2) µ(x, y) = 1, for all x ≤ P � = − µ(x, z), for all x < y in P. x⊇z ˆ 0. We need to The assertion is clearly true for rk(y) = 0, when y = ˆ show that � (−1)rk(x) = 0. (5) x⊇y ⎜ � The number of x such that x → y and rk(x) = i is ki , so (5) is equivalent to the well-known identity (easily proved by substituting q = −1 in the binomial expansion ⎜ � �k of (q + 1)k ) i=0 (−1)i ki = 0 for k > 0. �