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Gröbner and diagonal bases in Orlik-Solomon type algebras. (English) Zbl 1114.05020
The Orlik-Solomon algebra of a matriod $$\mathcal{M}$$ on the ground set $$\{1,2,\dots, n\}$$ is the quotient of the exterior algebra on the points by the ideal $$\mathfrak{I}(\mathcal{M})$$ generated by the circuits of $$\mathcal{M}$$. The isomorphism between the Orlik-Solomon algebra of a complex matroid and the complement of a complex arrangement of hyperplanes was established by P. Orlik and L. Solomon [Math. Ann. 261, 339–357 (1982; Zbl 0491.51018)].
In the article under review a generalization of Orlik-Solomon algebras, called $$\chi$$-algebra, is considered. Examples of $$\chi$$-algebras also include Orlik-Solomon-Terao algebras [P. Orlik and H. Terao, Nagoya Math. J. 134, 65–73 (1994; Zbl 0801.05019)] and Cordovil algebras [R. Cordovil, Discrete Comput. Geom. 27, No. 1, 73–84 (2001; Zbl 1016.52014)]. After the definition of $$\chi$$-algebras and the introduction of examples, special bases of the $$\chi$$-algebra are obtained from the “no broken circuit” sets of $$\mathcal{M}$$ and corresponding basis for the ideal $$\mathfrak{I}_{\chi} (\mathcal{M})$$. Furthermore, the reduced Gröbner basis for $$\mathfrak{I}_{\chi} (\mathcal{M})$$ is constructed. Finally, following A. Szenes [Int. Math. Res. Not. 1998, No. 18, 937–956 (1998; Zbl 0968.11015)], the so called diagonal basis for the $$\chi$$-algebra is defined and constructed. The results are illustrated by a small six point example.
##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 14F40 de Rham cohomology and algebraic geometry