Real Hyperplane Arrangement Research Articles

The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

The present paper is the first in a series of four dealing with a mapping, introduced by the present authors, from orientations to spanning trees in graphs, from regions to simplices in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). This mapping is actually defined for ordered oriented matroids. We call it the active orientation-to-basis mapping, in reference to an extensive use of activities, a notion depending on a linear ordering, first introduced by W.T. Tutte for spanning trees in graphs. The active mapping, which preserves activities, can be considered as a bijective generalization of a polynomial identity relating two expressions–one in terms of activities of reorientations, and the other in terms of activities of bases–of the Tutte polynomial of a graph, a hyperplane arrangement or an oriented matroid. Specializations include bijective versions of well-known enumerative results related to the counting of acyclic orientations in graphs or of regions in hyperplane arrangements. Other interesting features of the active mapping are links established between linear programming and the Tutte polynomial. We consider here the bounded case of the active mapping, where bounded corresponds to bipolar orientations in the case of graphs, and refers to bounded regions in the case of real hyperplane arrangements, or of oriented matroids. In terms of activities, this is the uniactive internal case. We introduce fully optimal bases, simply defined in terms of signs, strengthening optimal bases of linear programming. An optimal basis is associated with one flat with a maximality property, whereas a fully optimal basis is equivalent to a complete flag of flats, each with a maximality property. The main results of the paper are that a bounded region has a unique fully optimal basis, and that, up to negating all signs, fully optimal bases provide a bijection between bounded regions and uniactive internal bases. In the bounded case, up to negating all signs, the active mapping is a bijection.

The equivariant Orlik–Solomon algebra

Given a real hyperplane arrangement A , the complement M ( A ) of the complexification of A admits an action of the group Z 2 by complex conjugation. We define the equivariant Orlik–Solomon algebra of A to be the Z 2 -equivariant cohomology ring of M ( A ) with coefficients in the field F 2 . We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik–Solomon algebra into the Varchenko–Gelfand ring of locally constant F 2 -valued functions on the complement M R ( A ) of A in R n . We also show that the Z 2 -equivariant homotopy type of M ( A ) is determined by the oriented matroid of A . As an application, we give two examples of pairs of arrangements A and A ′ such that M ( A ) and M ( A ′ ) have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik–Solomon algebra.

Hyperplane arrangements and K-theory

We study the Z 2 -equivariant K-theory of M ( A ) , where M ( A ) is the complement of the complexification of a real hyperplane arrangement, and Z 2 acts on M ( A ) by complex conjugation. We compute the rational equivariant K- and KO-rings of M ( A ) , and we give two different combinatorial descriptions of a subring Line ( A ) of the integral equivariant KO-ring, where Line ( A ) is defined to be the subring generated by equivariant line bundles.

Properties of the residual circle action on a hypertoric variety

We consider an orbifold X obtained by a Kahler reduction of C n , and we define its hyperkahler analogue M as a hyperkahler reduction of T*C n ≅ H n by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkahler orbifold, as defined in Bielawski and Dancer, 2000, and further studied by Konno and by Hausel and Sturmfels. The variety M carries a natural action of S 1 , induced by the scalar action of S 1 on the fibers of T*C n . In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z 2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, real hyperplane arrangement H, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z 2 -equivariant cohomology of the complement of the complexification of H, where Z 2 acts by complex conjugation.

Faces of a Hyperplane Arrangement Enumerated by Ideal Dimension, with Application to Plane, Plaids, and Shi

The ideal dimension of a real affine set is the dimension of the intersection of its projective topological closure with the infinite hyperplane. We obtain a formula for the number of faces of a real hyperplane arrangement having given dimension and ideal dimension. We apply the formula to the plane, to plaids, which are arrangements of parallel families in general position, and to affinographic arrangements. We compare two definitions of ideal dimension and ask about a complex analog of the enumeration.

Geometric relationship between cohomology of the complement of real and complexified arrangements

Let A R be a real hyperplane arrangement and let A C be its complexification. Let M R and M C be the respective complements. Then M R is the disjoint union of convex chambers whose number is given by its only Betti number, b 0(M R ) . A real arrangement and its complexification satisfy the M-property: b 0(M R )=∑ qb q(M C ) , the number of chambers in M R equals the sum of the Betti numbers of M C . The no-broken-circuit set, nbc , is a field independent combinatorial object. It has been used to label a basis for H ∗(M C ) but not to label the chambers of M R in a way that makes the M-property explicit. In this paper we use the nbc set to label a combinatorial object in the nerve of the arrangement, which is field independent. This allows for simultaneous choices of nbc bases in H ∗(M R ) and H ∗(M C ) . We also explore the geometrical connections between these bases.

The Orlik–Solomon complex and Milnor fibre homology

Fix a real hyperplane arrangement, and let F be the Milnor fibre of its complexified defining polynomial. We consider a filtration of the homology of F that arises from the algebraic monodromy, using integer or Z/p Z coefficients. We compare it with the cohomology of the Orlik–Solomon algebra, over Z or Z/p Z , respectively, with respect to a suitable “Koszul” boundary map, and find isomorphisms in certain cases. This continues work by various authors in comparing the cohomology of certain local systems on a hyperplane complement with that of the Orlik–Solomon algebra.

Descent algebras, hyperplane arrangements, and shuffling cards

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types A A , B B , C C , H 3 H_3 , and rank two groups but not always. Both notions have the same simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W W is crystallographic and the relevant parameter is a good prime. The hyperplane viewpoint suggests deep connections with Lie theory and leads to a notion of riffle shuffling for arbitrary real hyperplane arrangements and oriented matroids.

The Dowling Transform of Subspace Arrangements

We define the Dowling transform of a real frame arrangement and show how the characteristic polynomial changes under this transformation. As a special case, the Dowling transform sends the braid arrangement An to the Dowling arrangement. Using Zaslavsky's characterization of supersolvability of signed graphs, we show supersolvability of an arrangement is preserved under the Dowling transform. We also give a direct proof of Zaslavsky's result on the number of chambers and bounded chambers in a real hyperplane arrangement.

Hanlon and Stanley's Conjecture and the Milnor Fibre of a Braid Arrangement

Let {\cal A} be a real arrangement of hyperplanes. Let B e B(q) be Varchenko's quantum bilinear form of {\cal A}, introduced [15], specialized so that all hyperplanes have weight q. B(q) is nonsingular for all complex q except certain roots of unity. Here, we examine the kernel of B at roots of unity in relation to the topology of the hyperplane singularity. We use Varchenko's work [16] to relate B(q) to a Salvetti complex for the Milnor fibration of {\cal A}. This paper's main result is specific to the arrangement of reflecting hyperplanes associated with the A_n − 1 root system. We use a geometric property of the Milnor fibre to resolve a conjecture due to Hanlon and Stanley regarding the {\Fraktur C}_n-module structure of the kernel of B(q) at certain roots of unity.

Integrable system of the heat kernel associated with logarithmic potentials

The heat kernel of a Sturm-Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrangement. The heat kernel satisfies an infinite-dimensional analog of the Gauss-Manin connection (i

The Toroidal Embedding Arising From an Irrational Fan

A toroidal embedding is defined which does not assume the fan consists of rational cones. For a rational fan, the toroidal embedding is the usual toric variety. If the fan is not rational, the toroidal embedding is in general a quasi-compact noetherian locally ringed space which is not a scheme. A divisor theory exists and a class group is defined. A second construction is also carried out which mimics the gluing construction of the usual toric variety, but which makes no reference to a lattice. The resulting scheme is separated but infinite dimensional. The Picard group is described in terms of the group of real valued locally linear support functions on the fan and the Brauer group is shown to be trivial. Many examples are given throughout the paper; in particular, it is shown that there is associated to a real hyperplane arrangement of full rank a toroidal embedding.

The Fundamental Group of the Complement of the Complexification of a Real Arrangement of Hyperplanes

Let A be an arrangement of hyperplanes (i.e., a finite set of 1-codimension vector subspaces) in Rd. We say that the linear ordering of the hyperplanes A,H1≺H2≺···≺Hn, is ashellabilityorder of A, if there is an oriented affine linelcrossing the hyperplanes of A on the given linear order. The intersection latticeL(A) is the set of all intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(Ac)≔Cd\\⋃{H⊗C:H∈A}. We will prove:Suppose that there are shellability orders H1≺H2≺···≺Hnand H′1≺′H′2≺′···≺′H′n, respectively, ofAandA′,such that the bijective map Hi→H′i, i∈[n]determines an isomorphism of the intersection lattices L(A)and L(A′).Then the fundamental groupsπ1(M(Ac))andπ1(M(A′c)) are isomorphic.

Random walks and hyperplane arrangements

Let $\mathscr{C}$ be the set of chambers of a real hyperplane arrangement. We study a random walk on $\mathscr{C}$ introduced by Bidigare, Hanlon and Rockmore. This includes various shuffling schemes used in computer science, biology and card games. It also includes random walks on zonotopes and zonotopal tilings. We find the stationary distributions of these Markov chains, give good bounds on the rate of convergence to stationarity, and prove that the transition matrices are diagonalizable. The results are extended to oriented matroids.

On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

We present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangement. The examples we consider include a family of complex subspace arrangements, which we call the divisor Dowling arrangement, whose intersection lattice generalizes that of the Dowling lattice. We also determine the characteristic polynomial of interpolations between subarrangements of the divisor Dowling arrangement, generalizing the work of Józefiak and Sagan.

Combinatorics of inductively factored arrangements

A real arrangement of hyperplanes is a finite family A of hyperplanes through the origin in a finite-dimensional real vector space V = R 1. A real arrangement A of hyperplanes is said to be factored if there exists a partition Π = ( Π 1, …, Π 1) of A into l disjoint subsets such that the Orlik-Solomon algebra of A factors according to this partition. A real arrangement A of hyperplanes is called inductively factored if it is factored and there exists a hyperplane H ϵ A such that the arrangement obtained by removing H from A and the arrangement on H consisting of all intersections of elements of A — H with H are both inductively factored. A chamber of A is a connected component of the complement of A . For a fixed base chamber, we may define a partial order on the set of chambers according to their combinatorial distances from the base chamber. Given an inductive factorization Π = ( Π 1,…( Π 1 and a base chamber C 0, we define the counting map of A with respect to C 0 as a morphism from the poset of chambers to the poset 0, 1,…,|Π 1| x·.x (0, 1,…,Π 1). We prove that, for a suitable base chamber, the counting map is a bijection, the poset of chambers is a lattice, and its rank-generating function has a nice factorization. We consider the dual decomposition of the sphere of V induced by A . We prove that, if A is inductively factored, then this cellular decomposition can be viewed as a decomposition of the boundary of the l-cube [0,|Π 1|]x·.x]0, |Π 1|[ by cubic cells.

The covers of a complexified real arrangement of hyperplanes and their fundamental groups

Let V be a finite dimensional real vector space and let A be an arrangement of hyperplanes in V. We denote by V C the complexification of V, by A C ={ H C = C ⊗ H| Hϵ A } the complexification of A , and by M( A ) the complement in V C of the union of all elements of A C . We associate with A an oriented graph Γ( A ) and some equivalonce relation ∽ in the set of (nonoriented) paths of Γ( A ). The pair (Γ( A ),∽) is called the oriented system associated with A . We introduce the notion of a cover of (Γ( A ),∽), from any cover of (Γ( A ),∽) we explicity construct a cover of M( A ), we prove that every cover of M( A ) can be obtained in this way and that a cover of ( Γ( A ),∽) is completely determined by the associated cover of M( A ). In particular, this method gives a construction of the universal cover of M( A ).

The deligne complex of a real arrangement of hyperplanes

Let V be a real vector space. An arrangement of hyperplanes in V is a finite family of hyperplanes of V through the origin. We say that is essential if ∩H ∊H = {0}

Some Almost Exceptional Arrangements

We introduce and study a family of real hyperplane arrangements that includes the reflection arrangements of types A and D as well as an infinite family of arrangements E′n (constructed by Orlik, Solomon, and Terao) related to the exceptional arrangements of type En, n ≤ 8. We investigate freeness (in the sense of Terao) of these arrangements, disproving the conjecture that the arrangements E′n are free in general.

Extension spaces of oriented matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings.We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.