Investigations On Hyperplane Arrangements And Matroids

This thesis studies two combinatorial aspects on hyperplane arrangements and matroids:classifications and bijections.The first is to classify the one-element extensions and co-extensions of a matroid representation,and classify the parallel translates of a linear hyperplane arrangement.In 1965,Crapo introduced one-element extension of a matroid and characterized all those extensions by the linear subclasses of its geometric lattice.In 1971,Rota introduced the derived matroid to study dependencies among all circuits of a given matroid.In 1989,Manin-Schechtman introduced discriminantal arrangement to classify all parallel translates of a given hyperplane.In this thesis,we found that the derived matroid and the discriminantal arrangement are closely related.Furthermore,the classification of all one-element coextensions of a matroid representation are characterized by its derived matroid,and the classification of all parallel translates of a given hyperplane arrangement is characterized by its discriminantal(or derived)arrangement.First introduced by Bixby-Coullard in 1988,the adjoint of a matroid representation turns out to be a dual concept of its derived matroid and in this thesis it will characterize the classification of all one-element extensions of a matroid representation.Based on these classifications,in this thesis we will further investigate some combinatorial invariants and obtain a semi-continuity on these invariants,including Whitney polynomials,characteristic polynomials,two kinds of Whitney numbers,face numbers,and region numbers etc..The second is to establish three bijections for extended Shi arrangement and Catalan arrangement in a uniform way by introducing a cubic matrix for each region of a hyperplane arrangement.In 1987,J.-Y.Shi introduced a classical hyperplane arrangement(Shi arrangment)and obtained the counting formula of its region numbers.As a celebrated algorithm on hyperplane arrangements obtained in 1996,the Pak-Stanley labelling established a one-to-one correspondence from the regions of(extended)Shi arrangements to(extended)parking functions.Most recently,Duarte-Guedes de Oliveira applied the Pak-Stanley labeling to the extended Catalan arrangement and obtained a bijection from its base regions to extended Dyck paths.In this thesis,we will introduce a cubic matrix for each region of a hyperplane arrangement and obtain three bijections on Shi and Catalan arrangements by reading data of the cubic matrix:(1)by reading the numbers of positive entries in its row slices,we will recover the Pak-Stanley labeling more directly;(2)by reading the numbers of positive entries in its column slices,we will obtain the Duarte and Guedes de Oliveira’s bijection;(3)by reading the positions of minimal positive entries in its column slices,we will establish a bijection from regions of the extended Shi arrangement to the extended labelled trees.