zbMATH — the first resource for mathematics

Orlik-Solomon type algebras. (English) Zbl 0984.52016
Summary: We introduce \(\chi\)-algebras, and show that a \(\chi\)-algebra has the NBC basis property. We also show that a certain ideal used in the construction has the so-called BC basis property. The Orlik-Solomon algebra of a matroid, the Orlik-Terao algebra of a set of vectors, and the Cordovil algebra of an oriented matroid are \(\chi\)-algebras. We define a new \(\chi\)-algebra from a set of vectors, close to the Orlik-Terao, and Cordovil algebras, but nevertheless different. Our proof provides a unified short and elementary proof of the NBC basis property for these algebras.

52C40 Oriented matroids in discrete geometry
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
Full Text: DOI
[1] Björner, A., The homology and shellability of matroids and geometric lattices, (), 226-283 · Zbl 0772.05027
[2] Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G., Oriented matroids, (1999), Cambridge University Press Cambridge
[3] R. Cordovil, A commutative algebra for oriented matroids, Discrete and Comp. Geom. (to appear.)
[4] Las Vergnas, M., Bases in oriented matroids, J. comb. theory, ser. B, 25, 283-289, (1978) · Zbl 0396.05005
[5] Orlik, P.; Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. math., 56, 167-189, (1980) · Zbl 0432.14016
[6] Orlik, P.; Solomon, L., Unitary reflection groups and cohomology, Invent. math., 59, 77-94, (1980) · Zbl 0452.20050
[7] Orlik, P.; Terao, H., Arrangements of hyperplanes, (1992), Springer-Verlag · Zbl 0757.55001
[8] Orlik, P.; Terao, H., Commutative algebras for arrangements, Nagoya math. J., 134, 65-73, (1992) · Zbl 0801.05019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.