Orlik-Solomon Algebra Research Articles

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.

Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.

Computations for Coxeter arrangements and Solomonʼs descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the graded components of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W. The refined conjecture relates the character above to a decomposition of the regular character of W related to Solomonʼs descent algebra of W. The refined conjecture has been proved for symmetric and dihedral groups, as well as for finite Coxeter groups of rank three and four. In this paper, we prove the conjecture for finite Coxeter groups of rank five and six. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik–Solomon characters of the groups considered.

Computations for Coxeter arrangements and Solomonʼs descent algebra: Groups of rank three and four

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group W afforded by the homogeneous components of its Orlik–Solomon algebra as sums of characters induced from linear characters of centralizers of elements of W. Our refined conjecture also relates the Orlik–Solomon characters above to the terms of a decomposition of the regular character of W related to the descent algebra of W. A consequence of our conjecture is that both the regular character of W and the character of the Orlik–Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of W, one for each conjugacy class of elements of W. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik–Solomon characters of these groups.

Isotropic subspaces of Orlik-Solomon algebras

We give a combinatorial characterization of isotropic subspaces in the Orlik-Solomon algebra of a hyperplane arrangement in terms of decorations of its intersection lattice. We then use this characterization to prove a result that relates these isotropic subspaces with linear systems supported on the arrangement, for arrangements with isolated non-normal crossings of a particular form.

Three-dimensional manifolds, skew-Gorenstein rings and their cohomology

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of the 3-manifold (bigger than - or maybe equal to - 11). We give also examples of 3-manifolds where the Ext-algebra of the cohomology ring of the fundamental group is not finitely generated.

Homological properties of Orlik–Solomon algebras

The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.

The chamber basis of the Orlik–Solomon algebra and Aomoto complex

We introduce a basis of the Orlik-Solomon algebra labeled by chambers, so called chamber basis. We consider structure constants of the Orlik-Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen's minimal complex from the Aomoto complex.

The Non-vanishing Cohomology of Orlik-Solomon Algebras

The cohomology of the complement of hyperplanes with coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements admitting weights for which the Orlik-Solomon algebra has non-vanishing cohomology, using decomposable relations arising from Latin hypercubes.

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.

Bases for certain cohomology representations of the symmetric group

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.

The Dimension of the Cohomology Groups of the Orlik–Solomon Algebras

The Orlik–Solomon algebra is a graded algebra defined by the partially ordered set of subspace intersections of the hyperplanes in an arrangement. Define the cohomology of an Orlik–Solomon algebra as that of the complex formed by its homogeneous components with the differential defined via multiplication by an element of degree one. The dimension of the cohomology of the Orlik–Solomon algebra in dimension one has been determined by Libgöber and Yuzvinsky. Using similar techniques, we study the dimension of the cohomology groups of the Orlik–Solomon algebra in higher dimensions under the special case where the element of degree one which defines the multiplication is concentrated under an element of the intersection lattice of codimension two. We provide computational methods for the dimension of the second cohomology group.

Cohomology of local systems coming fromp-adic hyperplane arrangements

For ap-adic hyperplane arrangement in a vector spaceV, we consider a local system of De Shalit on the Bruhat-Tits building ofPGL(V). We express this local system in terms of Orlik-Solomon algebras, and calculate its cohomology in the case where the arrangement is finite.

On Matroids and Orlik-Solomon Algebras

In this paper we axiomatize combinatorics of arrangements of affine hyperplanes, which is a generalization of matroids, called quasi-matroids. We show that quasi-matroids are equivalent to pointed matroids. On the other hand, the Orlik-Solomon (OS) algebra of a quasimatroid can be constructed. We prove that the OS algebra of a quasi-matroid is isomorphic to the direct image of the OS algebra of a matroid by the linear derivation.

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.

A note on Tutte polynomials and Orlik–Solomon algebras

Let A C ={H 1,…,H n} be a (central) arrangement of hyperplanes in C d and M( A C ) the dependence matroid of the linear forms {θ H i ∈( C d) ∗: Ker(θ H i )=H i} . The Orlik–Solomon algebra OS( M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra OS( M( A C )) is isomorphic to the cohomology algebra of the manifold M= C d⧹⋃ H∈ A C H . The Tutte polynomial T M (x,y) is a powerful invariant of the matroid M . When M( A C ) is a rank 3 matroid and the θ H i are complexifications of real linear forms, we will prove that OS( M) determines T M (x,y) . This result partially solves a conjecture of Falk.

Quadratic Orlik-Solomon algebras of graphic matroids

In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.

Hyperplane arrangements and linear strands in resolutions

The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field C \mathbf {C} over the Orlik-Solomon algebra.

Bases in Orlik–Solomon Type Algebras

Let M be a matroid on [ n ] and E be the graded algebra generated over a field k generated by the elements 1, e1,⋯ , en. Let (M) be the ideal of E generated by the squares e12,⋯ , en2, elements of the form eiej+aijejeiand ‘boundaries of circuits’, i.e., elements of the form ∑χjei1⋯eij−1eij+1⋯eim, with χj∈k and ei1,⋯ , eima circuit of the matroid with some special coefficients. The χ -algebra A(M) is defined as the quotient of E by (M). Recall that the class of χ -algebras contains several studied algebras and in first place the Orlik–Solomon algebra of a matroid. We will essentially construct the reduced Gröbner basis of (M) for any term order and give some consequences.

Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

Let G be a matroid on ground set A. The Orlik–Solomon algebra A(G) is the quotient of the exterior algebra E on A by the ideal I generated by circuit boundaries. The quadratic closure [formula](G) of A(G) is the quotient of E by the ideal generated by the degree-two component of I. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure [formula](G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math.28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G).The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π,1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π,1) or for A to be free.