António Guedes de Oliveira

Algebraic Varieties Characterizing Matroids and Oriented Matroids

Abstract Focusing on the interplay between properties of the Grassmann variety and properties of matroids and oriented matroids, this paper brings forward new algebraic methods in the theory of matroids and in the theory of oriented matroids; we introduce new algebraic varieties characterizing matroids and oriented matroids in the most general case. This new concept admits a systematic study of matroids and oriented matroids by using additional methods from calculus, algebra, and stochastics. An interesting new insight when the matroid variety over GF 2 and the chirotope variety over GF 3 are used shows that oriented matroids and matroids differ exactly by the underlying field. We investigate also a chirotope variety over R , its dimension, and its relation to the Grassmann variety. To find efficient algorithms in computational synthetic geometry, a crucial step lies in finding a small number of conditions for defining oriented matroids. Our new algebraic framework yields new results and straightforward proofs in this direction.

A note on the fundamental group of the Salvetti complex determined by an oriented matroid

Abstract LetMC(A) be the complexification of the complement of the hyperplanes of an arrangementAinRd. In [13], Salvetti constructed a regular finite CW complexX ⊂MC(A) homotopic to this space. The definition of this complex is essentially based on the structure of the oriented matroid determined byA, and can be extended similarly to other oriented matroids. In this note, we prove two theorems related to decompositions of the fundamental group of this generalised Salvetti complex.

Partitions of a finite Boolean lattice into intervals

Related to activities in matroids, J.E. Dawson introduced a construction that leads to partitions of the Boolean lattice of parts of a set into intervals. In this paper we characterize explicitly the partitions of a Boolean lattice into intervals that arise from this construction, and we prove that the construction is essentially unique.

Intersection and Linking Numbers in Oriented Matroids

Abstract We introduce for oriented matroids a generalization of the concepts of intersection and linking numbers in Euclidean space, with most of their main properties (see Wu). As an application, we reprove a result of Brehm in a slightly extended form.

On the Adjugate of a Matrix

1. S. Lang, Algebra 3rd ed., Addison-Wesley Publishing Company, New York, 1993. 2. J.-M. Pan, The order of the automorphism group of finite abelian group, J. Yunnan Univ. Nat. Sci. 26 (2004) 370–372. 3. A. Ranum, The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group, Trans. Amer. Math. Soc. 8 (1907) 71–91. 4. K. Shoda, Uber die Automorphismen einer endlischen Abelschen Gruppe, Math. Ann. 100 (1928) 674– 686.1. S. Lang, Algebra 3rd ed., Addison-Wesley Publishing Company, New York, 1993. 2. J.-M. Pan, The order of the automorphism group of finite abelian group, /. Yunnan Univ. Nat. Sei. 26 (2004)370-372. 3. A. Ranum, The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group, Trans. Amer. Math. Soc. 8 (1907) 71-91. 4. K. Shoda, ?ber die Automorphismen einer endlischen Abelschen Gruppe, Math. Ann. 100 (1928) 674 686.

Hyperplane arrangements between Shi and Ish

Abstract We introduce a new family of hyperplane arrangements in dimension n ≥ 3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the Pak-Stanley labelling. In addition, we characterise the Pak-Stanley labels of the regions of this family of hyperplane arrangements.

Between Shi and Ish

Abstract We introduce a new family of hyperplane arrangements in dimension n ≥ 3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – which can be bijectively labeled with the Pak–Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.

A generalized Desargues configuration and the pure braid group

Abstract In this paper, a configuration with n = ( 2 d ) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of ( 5 d ) such configurations if d > 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz-Zariski theorem implied by results of Salvetti that the fundamental group π of the complexification of its line arrangement is Artins pure (or coloured) braid group on d strands. In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of π based on this particular configuration.

On the Steinitz exchange lemma

A set of axioms of defining a matroid in terms of its bases is given by the Steinitz exchange lemma. In this paper, we show these axioms are not independent, and find a subcollection defining the same structure.A special motivation is given by the Graβmann variety and by oriented matroids, where we present improved versions of known results.