Broken circuit complexes and hyperplane arrangements

Abstract

We study Stanley-Reisner ideals of broken circuit complexes and characterize those ones admitting linear resolutions or being complete intersections. These results will then be used to characterize hyperplane arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for a matroid with a complete intersection broken circuit complex, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.

Graded Linearity of Stanley–Reisner Ring of Broken Circuit Complexes

“…is work is based on the results of [3,13] about several combinatorial characterizations of linear resolution property for the Stanley-Reisner ring of broken circuit complexes. e main theorem that is proven in that work is that of characterization for the decomposition of the broken circuit complex of a matroid when the associated Stanley-Reiner ring has a linear resolution.…”

“…e characterization of this property for Orlik-Terao ideals of arrangement of hyperplanes can be considered as a special case of having general linear resolution mentioned above. In this case, the Orlik-Terao ideal has a 2-linear resolution [3]. In this regard, the combinatorial identification of the Koszul property for algebras associated with matroids and hyperplane arrangements is one of the significant and crucial challenging problems in this area [5-7, 19, 20].…”

“…An interesting example is the Stanley-Reisner ideal I < (X) of the broken circuit complex of matroid X. e Stanley-Reisner ideal I < (X) is generated by monomials on the broken circuits of X. e broken circuit complex of an ordered matroid X is a shellable complex, where it shows the Stanley-Reisner ring A/I < (X) is Cohen-Macaulay [1]. e linearity property, in this case, can be translated into a combinatorial decomposition of the matroid in terms of uniform matroids (see [3]) in the form…”

“…Another important concept is the entire intersection property of Orlik-Solomon algebra and the Orlik-Terao ideal of hyperplane arrangements [3,[5][6][7].…”

“…ere appear several characterizations of this property in [3] concerning the linearity of the corresponding ideals. We express the generalizations of these theorems to the graded linearity as a natural application to our results.…”