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Gr¨obner and diagonal bases in Orlik-Solomon type algebras

Author
Cordovil, Raúl

Forge, David

Full text
http://revistas.ufro.cl/ojs/index.php/cubo/article/view/1539
Abstract
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal 𝔍(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, Andr´as 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 Gr¨obner bases of 𝔍χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an “iterative residue formula” introduced by Szenes.
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Artes, Arquitectura y UrbanismoCiencias Agrarias, Forestales y VeterinariasCiencias Exactas y NaturalesCiencias SocialesDerechoEconomía y AdministraciónFilosofía y HumanidadesIngenieríaMedicinaMultidisciplinarias
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Universidad de ChileUniversidad Católica de ChileUniversidad de Santiago de ChileUniversidad de ConcepciónUniversidad Austral de ChileUniversidad Católica de ValparaísoUniversidad del Bio BioUniversidad de ValparaísoUniversidad Católica del Nortemore

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