Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 63, No. 3, pp. 363-374 (2006)

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

Raul Cordovil and David Forge

Departamento de Matemática, Instituto Superior Técnico,
Av. Rovisco Pais, 1049-001 Lisboa -- PORTUGAL
E-mail: cordovil@math.ist.utl.pt
URL: http://www.math.ist.utl.pt/\ rcordov
Laboratoire de Recherche en Informatique UMR 8623,
Batiment 490, Université Paris Sud, 91405 Orsay Cedex -- FRANCE
E-mail: forge@lri.fr
URL: http://www.lri.fr/\ forge

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

Keywords: arrangement of hyperplanes; matroid; Orlik--Solomon algebra; biased graphs.

Classification (MSC2000): 05B35, 05Cxx, 14F40, 32S22.

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Electronic version published on: 7 Mar 2008.

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