Central Hyperplane Arrangement Research Articles

Inductive and divisional posets

AbstractWe call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so‐called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. Our first main result is that every divisional poset is factorable. Our second main result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (Adv. in Math. 52 (1984) 248–258), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third main result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type , or with respect to the root lattice is inductive.

Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b -functions and obtain a criterion to guarantee that roots of b -functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b -function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable n / d conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.

The Facial Weak Order on Hyperplane Arrangements

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.

A linear optimization oracle for zonotope computation

A class of counting problems asks for the number of regions of a central hyperplane arrangement. By duality, this is the same as counting the vertices of a zonotope. Efficient algorithms are known that solve this problem by computing the vertices of a zonotope from its set of generators. Here, we give an efficient algorithm, based on a linear optimization oracle, that performs the inverse task and recovers the generators of a zonotope from its set of vertices. We also provide a variation of that algorithm that allows to decide whether a polytope, given as its vertex set, is a zonotope and when it is not a zonotope, to compute its greatest zonotopal summand.

The Varchenko Determinant for Apartments

Varchenko introduced a distance function on chambers of hyperplane arrangements that he called quantum bilinear form. That gave rise to a determinant indexed by chambers whose entry in position (C, D) is the distance between C and D: that is the Varchenko determinant. He showed that that determinant has a nice factorization. Later, Aguiar and Mahajan defined a generalization of the quantum bilinear form, and computed the Varchenko determinant given rise by that generalization for central hyperplane arrangements and their cones. This article takes inspiration from their proof strategy to compute the Varchenko determinant given rise by their distance function for apartment of hyperplane arrangements. Those latter are in fact realizable conditional oriented matroids.

Incidence geometry in a Weyl chamber I: GLn

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gln restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five-dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U(n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.

The diffeomorphism type of small hyperplane arrangements is combinatorially determined

Abstract It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness, shown by means of symbolic computation, implies the desired result.

Intersection probabilities and kinematic formulas for polyhedral cones

For polyhedral convex cones in $${\mathbb{R}^d}$$ , we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.

Dimensional Reduction for Generalized Continuum Polymers

The Brydges-Imbrie dimensional reduction formula relates the pressure of a $d$-dimensional gas of hard spheres to a model of $(d+2)$-dimensional branched polymers. Brydges and Imbrie's proof was non-constructive and relied on a supersymmetric localization lemma. The main result of this article is a constructive proof of a more general dimensional reduction formula that contains the Brydges--Imbrie formula as a special case. Central to the proof are invariance lemmas, which were first introduced by Kenyon and Winkler for branched polymers. The new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to M\'esz\'aros and Postnikov. Several applications are presented, notably dimensional reduction formulas for (i) non-spherical bodies and (ii) for corrections to the pressure due to symmetry effects.

On Segre Numbers of Homogeneous Map Germs

Segre numbers and Segre cycles of ideals were independently introduced by Tworzewski, by Achilles and Manaresi and by Gaffney and Gassler. They are generalization of the Lê numbers and Lê cycles, introduced by Massey. In this article we give Lê-Iomdine type formulas for these cycles and numbers of arbitrary ideals. As a consequence we give a Plücker type formula for the Segre numbers of ideals generated by weighted homogeneous functions, in terms of their weights and degree. As an application of these results, we compute, in a purely combinatorial manner, the Segre numbers of the ideal which defines the critical loci of a map germ defined by a sequence of central hyperplane arrangements in $\mathsf{C}^{n+1}$.

Odd Khovanov homology for hyperplane arrangements

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincaré polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletion–restriction triples and dg-algebra structures. We also consider signed hyperplane arrangements, and generalize the odd Khovanov homology of Ozsváth–Rasmussen–Szabó from link projections to signed arrangements. We define hyperplane Reidemeister moves which generalize the usual Reidemeister moves from framed link projections to signed arrangements, and prove that the chain homotopy type associated to a signed arrangement is invariant under hyperplane Reidemeister moves.

Lattices of regular closed subsets of closure spaces

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg (P, φ), extending the poset Clop (P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg (P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg (P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,• Reg (P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.• If Reg (P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.• Clop (P, φ) is a lattice if and only if every regular closed set is clopen.The extended permutohedron R (G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R (G) and Reg S are bounded homomorphic images of free lattices.

Branched Polymers and Hyperplane Arrangements

We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie (Ann Math, 158:1019–1039, 2003), and Kenyon and Winkler (Am Math Mon, 116(7):612–628, 2009) to any central hyperplane arrangement $$\mathcal{A }$$ . The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement $$\mathcal{A }$$ is expressed through the value of the characteristic polynomial of $$\mathcal{A }$$ at 0. We give a more general definition of the space of branched polymers, where we do not require connectivity, and introduce the notion of q-volume for it, which is expressed through the value of the characteristic polynomial of $$\mathcal{A }$$ at $$-q$$ . Finally, we relate the volume of the space of branched polymers to broken circuits and show that the cohomology ring of the space of branched polymers is isomorphic to the Orlik–Solomon algebra.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Freeness and multirestriction of hyperplane arrangements

AbstractGeneralizing a result of Yoshinaga in dimension three, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

The Monodromy Conjecture for hyperplane arrangements

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: −n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in C n .

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ℂ2 with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ℂ*: the charged trigonometric Calogero-Moser system. When the particles lie on S1 ⊂ ℂ*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.

Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices

For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.

Affine and Toric Hyperplane Arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.