Gröbner and diagonal bases in Orlik-Solomon type algebras.

*(English)*Zbl 1114.05020The 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.

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.

Reviewer: Brigitte Servatius (Worcester)

##### 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 |