Orlik-Solomon Algebra Research Articles

On the homology of the noncrossing partition lattice and the Milnor fibre

Let L be the noncrossing partition lattice associated to a finite Coxeter group W. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of L. We define a multiplicative structure on the Whitney homology of L in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of W, the Milnor fibre of the discriminant of W, the hyperplane complement of W and the Artin group of type W, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.

Equivariant Log Concavity and Representation Stability

AbstractWe expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{s}\mathfrak{l}_n$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree.

A Leray Model for the Orlik–Solomon Algebra

Abstract We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blow-up data, we give a presentation for a bigraded (commutative) differential graded algebra. If the matroid is realizable over $\mathbb {C}$, this is the familiar Morgan model for a hyperplane arrangement complement, embedded in a blowup of projective space. In general, we obtain a cdga that interpolates between the Chow ring of a matroid and the Orlik–Solomon algebra. Our construction can also be expressed in terms of sheaves on combinatorial blowups of geometric lattices. As a key technical device, we construct a monomial basis via a Gröbner basis for the ideal of relations. Combining these ingredients, we show that our algebra is quasi-isomorphic to the classical Orlik–Solomon algebra of the matroid.

Stability phenomena for resonance arrangements

We prove that the i th i^\text {th} graded pieces of the Orlik–Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS o p \operatorname {FS}^{\mathrm {op}} -module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.

A Study of The Cohomology of The Orlik-Solomon Algebra As a Free Module For a Generic Arrangement

This paper is devoted to study the cohomology of the Orlik-Solomon algebra A(A) as a free module for an -generic -arrangement and , . In particular, the dimension of has been determined for every .

Universally Koszul and initially Koszul properties of Orlik–Solomon algebras

Let [Formula: see text] be a field with [Formula: see text] and [Formula: see text] an exterior algebra over [Formula: see text] with a standard grading [Formula: see text]. Let [Formula: see text] be a graded algebra, where [Formula: see text] is a graded ideal in [Formula: see text]. In this paper, we study universally Koszul and initially Koszul properties of [Formula: see text] and find classes of ideals [Formula: see text] which characterize such properties of [Formula: see text]. As applications, we classify arrangements whose Orlik–Solomon algebras are universally Koszul or initially Koszul. These results are related to a long-standing question of Shelton–Yuzvinsky [B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997) 477–490].

Koszulness and supersolvability for Dirichlet arrangements

We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet arrangements that are combinatorially distinct from these other four classes.

Relative cohomology of bi-arrangements

A bi-arrangement of hyperplanes in a complex affine space is data of two sets of hyperplanes along with a coloring information on strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from notion of motivic period. More generally, we suggest systematic study of motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute structure of these motives. Our main object is Orlik-Solomon bi-complex of a bi-arrangement, which generalizes Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex, which generalizes classical facts involving Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.

A Deformation of the Orlik-Solomon Algebra

A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

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.

Counting resolutions of symplectic quotient singularities

Let ${\rm\Gamma}$ be a finite subgroup of $\text{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V/{\rm\Gamma}$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero–Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik–Solomon algebra naturally associated to the Calogero–Moser deformation. This dimension is explicitly calculated for all groups ${\rm\Gamma}$ for which it is known that $V/{\rm\Gamma}$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.

Supersolvability and the Koszul property of root ideal arrangements

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.

Lattice and Heegaard Floer Homologies of Algebraic Links

We compute the Heegaard Floer link homology of algebraic links in terms of the multi-variate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a generalized version of the Orlik–Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincaré series of the singularity.

The Orlik-Solomon model for hypersurface arrangements

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.

Resonant bands, Aomoto complex, and real 4-nets

Introduction The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. We develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist. Let us fix some notation: k ∈ Z, k ≥ 3; K a field (generally, R or C) and KP2 the projective plane; A = {H0, . . . ,Hn} a line arrangement in KP2; A = {H1, . . . ,Hn} the affine line arrangement in K2 = KP2 H0 obtained from A; AF2(A) the Orlik-Solomon algebra of A over F2 generated by the symbols e1, . . . , en; For S ⊂ A, consider e(S) := ∑ Hi∈S ei ∈ A 1 F2(A) and η0 := e(A) = ∑n i=1 ei. Definition (k-nets) A supports a k -net structure if and only if there exist a partition A = A1 t · · · t Ak and a finite set of points X ⊂ KP2 such that For all i 6= j , if H ∈ Ai and H ′ ∈ Aj, then H ∩ H ′ ∈ X ; For all p ∈ X and for all i = 1, . . . , k , there exists a unique H ∈ Ai such that p ∈ H. Known facts If k ≥ 5 there does not exist any k -net; There exist infinitely many 3-nets; The Hesse arrangement is the only known 4-net. Theorem 2 (Papadima-Suciu) Consider S A. Then e(S) ∧ η0 = 0 if and only if ∀p ∈ K2 one of the following is satisfied: if |Ap| is odd, then |Ap| = |Sp|; if |Ap| is even then |Sp| is even, where Ap := {H ∈ A | p ∈ H}. Hesse arrangement Consider fλ := λ(x3 + y3 + z3)− 3xyz ∈ C[x, y , z] with λ ∈ C. Now, fλ factors into linear factors if and only if λ ∈ {0, 1, ω, ω2}, where ω = e2πi/3. Consider now A1 := {f0 = 0}, A2 := {f1 = 0}, A3 := {fω = 0} and A4 := {fω2 = 0}. Define the complex arrangement A = A1 t A2 t A3 t A4. This partition give as a 4-net structure on A. Theorem 1 (Papadima-Suciu) Let Q be a defining equation of A over C. Consider F := {(x, y , z) ∈ C3 | Q(x, y , z) = 1} the Milnor fiber of A. If A supports a k -net structure then dim H1(F )e2πi/k ≥ k − 2. Corollary If A has a 4-net structure, then dim H(AF2(A), η0) ≥ 2.

Degeneration of Orlik–Solomon algebras and Milnor fibers of complex line arrangements

We give a vanishing theorem for the monodromy eigenspaces of the Milnor fibers of complex line arrangements. By applying the modular bound of the local system coho- mology groups given by Papadima-Suciu, the result is deduced from the vanishing of the cohomologyofcertainAomotocomplexoverfinitefields.Inordertoprovethis,weintroduce degeneration homomorphisms of Orlik-Solomon algebras.

Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight

In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the p-th graded component of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.

Cohomology of Coxeter arrangements and Solomon’s descent algebra

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group W W and relate it to the descent algebra of W W . As a result, we claim that both the group algebra of W W and the Orlik-Solomon algebra of W W can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W W . We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair ( W , W L ) (W, W_L) , where W W is arbitrary and W L W_L is a parabolic subgroup of W W , all of whose irreducible factors are of type A A .

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.

The Orlik-Solomon algebra and the supersolvable class of arrangements

According to the powerful geometric properties of the hypersolvable order on the hyperplanes of a supersolvable arrangement, we introduced a sufficient condition on the Orlik-Solomon algebra for any central arrangement to have supersolvable analogue and we showed this condition as a necessary condition (not sufficient) on the Orlik-Solomon algebra for any central arrangement to be K(π; 1). Finally as an illustration of our result, we produce to the Orlik-Solomon algebra for the complexification of the Coxeter arrangement of type Ar and Br, for r ≥ 3, a structure by their supersolvable partitions analogues.