Combinatorial and Algebraic Structure in Orlik–Solomon Algebras (original) (raw)

A Note on the Orlik–Solomon Algebra

European Journal of Combinatorics, 2001

Let M = M(E) be a matroid on a linear ordered set E. The Orlik-Solomon Z-algebra OS(M) of M is the free exterior Z-algebra on E, modulo the ideal generated by the circuit boundaries. The Z-module OS(M) has a canonical basis called 'no broken circuit basis' and denoted nbc. Let e X = e i , e i ∈ X ⊂ E. We prove that when e X is expressed in the nbc basis, then all the coefficients are 0 or ±1. We present here an algorithm for computing these coefficients. We prove in appendix a numerical identity involving the dimensions of the algebras of Orlik-Solomon of the minors of a matroid and its dual.

Diagonal Bases in Orlik-Solomon Type Algebras

Annals of Combinatorics, 2003

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal ℑ(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called χ-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, András Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to the χ-algebras. We give a survey of the results obtained by the authors concerning the construction of Gröbner bases of ℑχ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an "iterative residue formula" introduced by Szenes.

Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

Advances in Applied Mathematics, 2002

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 A(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 nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(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. These results generalize to the degree r closure of A(G).

Groebner and diagonal bases in Orlik-Solomon type algebras

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to X-algebras. We give a survey of the results obtained by the authors concerning the construction of Groebner bases of I(M) and diagonal bases of Orlik-Solomo...

Gröbner and diagonal bases in Orlik-Solomon type algebras

2005

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal (M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called χ-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, András Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to the χ-algebras. We give a survey of the results obtained by the authors concerning the construction of Gröbner bases of χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an "iterative residue formula" introduced by Szenes.

An Orlik-Solomon type algebra for matroids with a fixed linear class of circuits

A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.

Line-closed matroids, quadratic algebras, and formal arrangements

2000

Let G be a matroid on ground set . The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra on by the ideal generated by circuit boundaries. The quadratic closure A̅(G) of A(G) is the quotient of by the ideal generated by the degree-two component of . We introduce the notion of set in G, determined by a linear order on , and show that the corresponding monomials are linearly independent in the quadratic closure A̅(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. These results generalize to the degree r closure of (G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for to be free and for the complement M of to be a K(π,1) space. Formality of 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 formalit...

A Commutative Algebra for Oriented Matroids

Discrete & Computational Geometry, 2002

Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K-algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: • Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik-Terao, which is the main tool of our proofs.

Quadratic Orlik-Solomon algebras of graphic matroids

Matemática contemporânea, 2003

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.

A note on Tutte polynomials and Orlik–Solomon algebras

European Journal of Combinatorics, 2003

Let A C = {H 1 ,. .. , Hn} 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 three 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 solves partially a conjecture of M. Falk.