Subspace Arrangements Research Articles

Lefschetz duality for local cohomology

Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence.

Intersection Theory of Polymatroids

Abstract Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of subspace arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids. We show that intersection numbers in the augmented Chow ring of a polymatroid are determined by a matching property known as the Hall–Rado condition, which is new even in the case of matroids.

Von Staudt constructions for skew-linear and multilinear matroids

This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results:Undecidability of several matroid representation problems over division rings. An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic \(0\). An example of a skew-linear matroid that is not multilinear.  Mathematics Subject Classifications: 05B35, 52B40, 14N20, 52C35, 20F10, 03D40Keywords: Matroids, division ring representations, subspace arrangements, \(c\)-arrange\-ments, multilinear matroids, von Staudt constructions, word problem, Weyl algebra, Baumslag-Solitar group

Resolutions of ideals of subspace arrangements

Given a collection of t subspaces in an n-dimensional vector space W we can associate to them t linear ideals in the symmetric algebra 𝒮(W∗). Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of t linear ideals is at most t. We show that analogous results hold when we work over the exterior algebra ⋀(W∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.

Generalized Virtual Polytopes and Quasitoric Manifolds

We develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the Bernstein–Kushnirenko theorem as well as Stanley–Reisner and Pukhlikov–Khovanskii type descriptions for the cohomology rings of generalized quasitoric manifolds.

Equivariant Burnside groups and representation theory

We apply the equivariant Burnside group formalism to distinguish linear actions of finite groups, up to equivariant birationality. Our approach is based on De Concini–Procesi models of subspace arrangements.

Resolution of ideals associated to subspace arrangements

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

On the rank of an A$A$‐hypergeometric D$D$‐module versus the normalized volume of A$A$

The rank of an $A$-hypergeometric $D$-module $M_A(\beta)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $\beta\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $\beta$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $\beta\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(\beta))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $\beta$ such that this ratio is at least $2$ is an affine subspace arrangement of codimension at least $3$.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

Homomorphic sensing of subspace arrangements

Homomorphic sensing is a recent algebraic-geometric framework that studies the unique recovery of points in a linear subspace from their images under a given collection of linear maps. It has been successful in interpreting such a recovery in the case of permutations composed by coordinate projections, an important instance in applications known as unlabeled sensing, which models data that are out of order and have missing values. In this paper, we provide tighter and simpler conditions that guarantee the unique recovery for the single-subspace case, extend the result to the case of a subspace arrangement, and show that the unique recovery in a single subspace is locally stable under noise. We specialize our results to several examples of homomorphic sensing such as real phase retrieval and unlabeled sensing. In so doing, in a unified way, we obtain conditions that guarantee the unique recovery for those examples, typically known via diverse techniques in the literature, as well as novel conditions for sparse and unsigned versions of unlabeled sensing. Similarly, our noise result also implies that the unique recovery in unlabeled sensing is locally stable.

Unitary representations of the Cherednik algebra: $$V^*$$-homology

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the $$V^*$$ -homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups.

Subspace Arrangements and Cherednik Algebras

Abstract The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category ${\mathscr{O}}$ is equivalent to knowledge of the Kazhdan–Lusztig character of the irreducible object (we use this observation in joint work with Fishel–Manosalva). We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the $k$-equals arrangement. In the case of the radical, we apply our results with Juteau together with an idea of Etingof–Gorsky–Losev to observe that the quotient is Cohen–Macaulay for positive choices of parameters. In the case of the socle (in cyclotomic type), we give an explicit vector space basis in terms of certain specializations of nonsymmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov.

A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids

We advocate a systematic study of continuous analogs of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples reviewed are an Euler formula for a class of “continuous convex polytopes” (conjectured by Kalai and Wigderson), a duality result for a class of “continuous matroids,” a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of G.-C. Rota), an exposition of the “homotopy complementation formula” for topological posets and its relation to the results of S. Kallel and R. Karoui about “weighted barycenter spaces,” and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria’s inequality) based on interpreting diagrams of spaces as continuous posets.

On a generalization of Solomon-Terao formula for subspace arrangements

We investigate in this paper a generalization of Solomon-Terao formula for central equidimensional subspace arrangements. We introduce generalized Solomon-Terao functions based on the Hilbert-Poincaré series of the modules of multi-logarithmic forms and logarithmic multi-residues. We show that as in the case of hyperplane arrangements, these Solomon-Terao functions are polynomial. We then prove that if the Solomon-Terao polynomial of the modules of multi-residues satisfies a certain property, then this polynomial is related to the characteristic polynomial of the subspace arrangement. In particular, we prove that this generalized Solomon-Terao formula holds for any line arrangement of any codimension.

Wonderful Models for Generalized Dowling Arrangements

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon (Trans. Amer. Math. Soc. 325(1), 1–37, 1991). In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components in the boundary, which provides a geometric realization of the nested set poset of this generalized Dowling lattice. It can be represented by a family of forests with leaves and labelings that depend on the triple (n,G,V ). We will study it from the enumerative point of view in the case when G is abelian.

Classifications of ℓ-Zero-Sumfree Sets


 
 
 The set of all ℓ-zero-sumfree subsets of ℤ/nℤ is a simplicial complex denoted by Δn, ℓ. We create an algorithm via defining a set of integer partitions we call (n,ℓ)-congruent partitions in order to compute this complex for moderately-sized parameters n and ℓ. We also theoretically determine Δn, ℓ for several infinite families of parameters, and compute the intersection posets and the characteristic polynomials of the corresponding coordinate subspace arrangements.
 
 
 

A rotation group whose subspace arrangement is not from a real reflection group

We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino and Singh.

Face numbers of Engström representations of matroids

A classic problem in the theory of matroids is to find a subspace arrangement, such as a hyperplane or pseudosphere arrangement, whose intersection poset is isomorphic to a prescribed geometric lattice. Engström gave an explicit construction for an infinite family of such arrangements, indexed by the set of finite regular CW complexes. In this paper, we compute the face numbers of these topological representations in terms of the face numbers of the indexing complexes and give upper bounds on the total number of faces in these objects. Moreover, for a fixed rank, we show that the total number of faces in the Engström representation corresponding to a codimension one homotopy sphere arrangement is bounded above by a polynomial in the number of elements of the matroid, whose degree is one less than the matroid’s rank.

Toric varieties of Loday's associahedra and noncommutative cohomological field theories

We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, coming from the fact that they admit several equivalent interpretations: as the toric varieties of Loday's realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini-Procesi wonderful models for certain subspace arrangements.

Representation stability on the cohomology of complements of subspace arrangements

We study representation stability in the sense of Church and Farb of sequences of cohomology groups of complements of arrangements of linear subspaces in real and complex space as S n -modules. We consider arrangements of linear subspaces defined by sets of diagonal equalities x i =x j and invariant under the action of S n which permutes the coordinates. We provide bounds for the point when stabilization occurs and an alternative proof of the fact that stabilization happens. The latter is a special case of very general stabilization results proved independently by Gadish and by Petersen; for the pure braid space the result is part of the work of Church and Farb. For the latter space, better stabilization bounds were obtained by Hersh and Reiner.