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.     

Between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension $n\ge 3$ that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – which can be bijectively labeled with the Pak–Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.     

On cohomology algebras of complex subspace arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

     

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form xj=xi+c are called affinographic. For an affinographic hyperplane arrangement in Rn, such as the Shi arrangement, we study the function f(m) that counts integral points in n[1,m] that do not lie in any hyperplane of the arrangement. We show that f(m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph.An application is to interval coloring in which the interval of available colors for vertex vi has the form [hi+1,m].A related problem takes colors modulo m; the number of proper modular colorations is a different piecewise polynomial that for large m becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.     

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.     

Fully Optimal Bases and the Active Bijection in Graphs,Hyperplane Arrangements,and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the active bijection between bounded regions and uniactive internal bases. The active bijec-tion is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations.     

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.     

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 .     

Completely reducible hypersurfaces in a pencil

We study completely reducible fibers of pencils of hypersurfaces on Pn and associated codimension one foliations of Pn. Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of n. Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements.     

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w∈Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups.A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.     

On duality gap in binary quadratic programming

We investigate in this paper the duality gap between the binary quadratic optimization problem and its semidefinite programming relaxation. We show that the duality gap can be underestimated by ${\xi_{r+1}\delta^2}$ , where ?? is the distance between {?1, 1} ⁿ and certain affine subspace, and ?? _r+1 is the smallest positive eigenvalue of a perturbed matrix. We also establish the connection between the computation of ?? and the cell enumeration of hyperplane arrangement in discrete geometry.     

Fundamental groups of blow-ups

Many examples of nonpositively curved closed manifolds arise as real blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W and if the blow-up locus is W-invariant, then the resulting manifold will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P. In other words, M admits a tiling with tile P. The universal covers of such examples yield tilings of whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these “mock reflection groups”, and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.     

On the division of space by topological hyperplanes

A topological hyperplane is a subspace of (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in is a finite set such that for any nonvoid intersection Y of topological hyperplanes in and any that intersects but does not contain Y, the intersection is a topological hyperplane in Y. (We also assume a technical condition on pairwise intersections.) If every two intersecting topological hyperplanes cross each other, the arrangement is said to be transsective. The number of regions formed by an arrangement of topological hyperplanes has the same formula as for arrangements of ordinary affine hyperplanes, provided that every region is a cell. Hoping to explain this geometrically, we ask whether parts of the topological hyperplanes in any arrangement can be reassembled into a transsective arrangement of topological hyperplanes with the same regions. That is always possible if the dimension is two but not in higher dimensions. We also ask whether all transsective topological hyperplane arrangements correspond to oriented matroids; they need not (because parallelism may not be an equivalence relation), but we can characterize those that do if the dimension is two. In higher dimensions this problem is open. Another open question is to characterize the intersection semilattices of topological hyperplane arrangements; a third is to prove that the regions of an arrangement of topological hyperplanes are necessarily cells; a fourth is whether the technical pairwise condition is necessary.     

Bruhat order, smooth Schubert varieties, and hyperplane arrangements

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincaré polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Addition–Deletion Theorems for factorizations of Orlik–Solomon algebras and nice arrangements

We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao’s celebrated addition–deletion theorem for free arrangements for the class of nice arrangements. This is a natural setting for the stronger property of an inductive factorization of a hyperplane arrangement by Jambu and Paris.In addition, we show that supersolvable arrangements are inductively factored and that inductively factored arrangements are inductively free. Combined with our addition–deletion theorem this leads to the concept of an induction table for inductive factorizations.Finally, we prove that the notions of factored and inductively factored arrangements are compatible with the product construction for arrangements.     

Deletion-restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A₀. Let A^′=A?{A₀} be the deletion of A₀ from A and A^″={A∩A₀|A∩A₀≠∅} be the restriction of A to A₀. Let M=V?_?A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M^′)→Hk(M)→Hk+1−codimR(A₀)(M^″)→0. In this paper, we determine conditions for when the triple (A,A^′,A^″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.     

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the Riemann hypothesis if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system A _{n – 1}. The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.