On Matroids and Orlik-Solomon Algebras

Abstract

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.

Group actions on semimatroids

“…The motivation for introducing these structures was, in both [2] and [24], the combinatorial study of affine hyperplane arrangements. In particular, keeping an eye on Example 1.6 below will help make the following definition plausible.…”

“…We initiate the study of actions of groups by automorphisms on semimatroids (for short "G-semimatroids"). Helpful intuition comes, once again, from the case of integer vectors, where the associated toric arrangement is covered naturally by a periodic affine hyperplane arrangement: here semimatroids, introduced by Ardila [2] (independently by Kawahara [24]), enter the picture as abstract combinatorial descriptions of finite arrangements of affine hyperplanes. In particular, we obtain the following results (see also Table 1 for a quick overview).…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…65. Let E n (u), u = 0, 1, be the Yokonuma-Hecke algebra, see, e.g., [124] and the literature quoted therein.…”

The Non-vanishing Cohomology of Orlik-Solomon Algebras

“…If ∂e λ = n j=1 λ j = 0 then e λ induces the complex ( For a generic weight λ, Yuzvinsky [15] showed the vanishing theorem: An arrangement A of hyperplanes in P ℓ has the underlying matroid M (A) = M with rank ℓ + 1 as a combinatorial structure. The cohomology of the complement of A is isomorphic to ∂ M (A(M )) (see [10] and [7]). If a weight λ = (λ i ) i∈A satisfies some generic condition, then the cohomology of the complement of A with the coefficients in the rank one local system associated to λ is isomorphic to H(∂ M (A(M )), e λ ) (see [5,14]).…”