Gröbner and Diagonal Bases in Orlik-solomon Type Algebras
نویسنده
چکیده
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 OrlikSolomon 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 OrlikSolomon-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.
منابع مشابه
Diagonal Bases in Orlik-solomon Type Algebras
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 basis”. We prove that this definition extends naturally to a large class of algebras, the so called χ-algebras. Our definitions make also use of an “iterative residue formula” based on the matroidal operation of co...
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