Hyperplane Arrangements Research Articles

Projective Geometry of Wachspress Coordinates

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

Higher holonomy maps for hyperplane arrangements

We develop a method to construct representations of the homotopy 2-groupoid of a manifold as a 2-category by means of Chen’s formal homology connections. As an application we describe 2-holonomy maps for hyperplane arrangements and discuss representations of the category of braid cobordisms.

Hyperplane Arrangements and Tensor Product Invariants

In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map. In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.

The Chow Form of a Reciprocal Linear Space

A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.

Bisecting measures with hyperplane arrangements

AbstractWe show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.

Solomon–Terao algebra of hyperplane arrangements

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon–Terao algebra $ST(\mathcal{A}, \eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $ST(\mathcal{A}, \eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon–Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $ST(\mathcal{A}, \eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials of $ST(\mathcal{A}, \eta)$ when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.

On the length of perverse sheaves on hyperplane arrangements

In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes.

Free hyperplane arrangements over arbitrary fields

In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over \(\mathbb {Q}\).

Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

In 1995, the first author introduced a multivariate generating function G that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that G is symmetric, he conjectured that G is Schur positive. We prove this conjecture by expanding G positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still Schur positive.We also discuss applications in the setting of hyperplane arrangements. We show that a certain specialization of G equals the Frobenius characteristic of the natural Sn-action on regions of the semiorder arrangement, which we then expand in terms of the Frobenius characteristics of Foulkes characters. We also construct an Sn-action on regions of the Linial arrangement using a set of trees studied by Bernardi, and subsequently compute the character of this action by employing Lagrange inversion. The resulting expression generalizes Postnikov's formula for the number of regions in the Linial arrangement. As a final application, we prove γ-nonnegativity for the distribution of the number of right edges over local binary search trees.

Cambrian triangulations and their tropical realizations

This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on ν-Tamari lattices and their tropical realizations. For any signature ε∈{±}n, we consider a family of ε-trees in bijection with the triangulations of the ε-polygon. These ε-trees define a flag regular triangulation Tε of the subpolytope conv(ei•,ej∘)|0≤i•<j∘≤n+1 of the product of simplices △{0•,…,n•}×△{1∘,…,(n+1)∘}. The oriented dual graph of the triangulation Tε is the Hasse diagram of the (type A) ε-Cambrian lattice of N. Reading. For any I•⊆{0•,…,n•} and J∘⊆{1∘,…,(n+1)∘}, we consider the restriction TI•,J∘ε of the triangulation Tε to the face △I•×△J∘. Its dual graph is naturally interpreted as the increasing flip graph on certain (ε,I•,J∘)-trees, which is shown to be a lattice generalizing in particular the ν-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of TI•,J∘ε as a polyhedral complex induced by a tropical hyperplane arrangement.

Electrical networks and hyperplane arrangements

This paper introduces Dirichlet arrangements, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize combinatorial results on graphic arrangements to Dirichlet arrangements, addressing characteristic polynomials and supersolvability in particular. We apply these results to visibility sets of order polytopes and fixed-energy harmonic functions on electrical networks.

Chern–Schwartz–MacPherson cycles of matroids

We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.

Computing Optimal Control of Cascading Failure in DC Networks

We consider discrete-time dynamics for cascading failure in DC networks whose map is a composition of failure rule with control actions. Supply–demand at the nodes is monotonically nonincreasing under admissible control. Under the failure rule, a link is removed permanently if its flow exceeds capacity. We consider finite horizon optimal control to steer the network from an arbitrary initial state, defined in terms of active link set and supply–demand at the nodes, to a feasible state, i.e., a state which is invariant under the failure rule. There is no running cost and the reward associated with a feasible terminal state is the associated cumulative supply–demand. We propose two approaches for computing optimal control. The first approach, geared toward tree reducible networks, decomposes the global problem into a system of coupled local problems, which can be solved to optimality in two iterations. When restricted to the class of one-shot control actions, the optimal solutions to the local problems possess a piecewise affine property, which facilitates analytical solutions. The second approach computes optimal control by searching over the reachable set, which is shown to admit an equivalent finite representation by aggregation of control actions leading to the same reachable active link set. An algorithmic procedure to construct this representation is provided by leveraging and extending tools for arrangement of hyperplanes and polytopes. Illustrative simulations, including showing the effectiveness of projection-based approximation algorithms, are also presented.

Optimal arrangements of hyperplanes for SVM-based multiclass classification

In this paper, we present a novel approach to construct multiclass classifiers by means of arrangements of hyperplanes. We propose different mixed integer (linear and non linear) programming formulations for the problem using extensions of widely used measures for misclassifying observations where the \textit{kernel trick} can be adapted to be applicable. Some dimensionality reductions and variable fixing strategies are also developed for these models. An extensive battery of experiments has been run which reveal the powerfulness of our proposal as compared with other previously proposed methodologies.

Action dimensions of some simple complexes of groups

The action dimension of a discrete group G is the minimum dimension of a contractible manifold that admits a proper G-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including (1) Artin groups, (2) graph products of groups and (3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.

Combinatorial analogs of topological zeta functions

In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function ZL(s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolutions of singularities, and the main result is that ZL(s) does not depend on this choice and depends only on L. Known properties of the topological zeta function provide a source of potential ℂ-realisability test for matroids.

Plus-one Generated and Next to Free Arrangements of Hyperplanes

AbstractWe introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, that is, plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus-one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show that a free arrangement is in fact surrounded by free or strictly plus-one generated arrangements. We also give several applications.

Multiple addition, deletion and restriction theorems for hyperplane arrangements

In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem in [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320]. Recently, the multiple version of the addition theorem was proved in [J. Eur. Math. Soc. 18 (2016), 1339–1348], called the multiple addition theorem (MAT), to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT to get MAT2 from the viewpoint of our new proof. Moreover, we introduce the restriction version, a multiple restriction theorem (MRT). Applications of MAT2, including the combinatorial freeness of the extended Catalan arrangements, are given.

Hyperplane arrangements in CoCoA

We introduce the package \textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.

Alternation acyclic tournaments

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a simple model for the normalized median Genocchi numbers.