On The Class Of Factored Arrangements

Abstract

The first main objective of the work was to create a combinatorial answer to an essential question; "How Terao generalization of the class of supersolvable arrangements preserved the tensor factorization of the O-S algebra?", by finding a relation among several bases of the O-S algebra. This was achieved in two parts. First, the class of factored arrangements was classified in two subclasses, the subclass of completely factored arrangements and the subclass of factored arrangement that not completely factored. Our classification criteria was, "the existence of an ordering  on the hyperplanes of a factored arrangement  such that the set of all monomials that related to the sections of a factorization  on  forms an NBC basis of the O-S algebra as a free module". The second part was, a comparison among the structures of the O-S complex, the NBC complex and the partition complex. In spite of, our classification criteria was failed of the second subclass of factored arrangement that not completely factored, the existence of a one to one correspondence between the set of all NBC bases of  and the set of all sections of a factorization  on , provides a tensor factorization fashion to the O-S algebra.

The second main aim of the work was to prove that our classification is compatible with the product construction for arrangements, by constructing the O-S complex, the NBC complex and the partition complex of the reducible factored arrangements. Finally, several illustrations and applications were indicated.

Keywords: Hyperplane arrangement, Supersolvable arrangement, factored (Nice) arrangement, Orlik-Solomon algebra, NBC module and Partition tensor module.