Combinatorial And Algebraic Aspects Of The Mixed Arrangement

Since B .Gr(?)nbaum first suggest the concept arrangement of hyperplane in 1971, many mathematicians of the world have made impressive contribution in the combinatorics, algebra and topology aspects of the hyperplane arrangement. As the natural extensive of the concept of the hyperplane arrangement, the concept of the subspace arrangement was suggested naturally after that. Here we have to mention the following theorem first, which was proved by M. Goresky and R. MacPhersonTheorem The homology of the complement C = Rⁿ\U_i=0^tAi is given bywhereΔis a function sending every poset to it order complex and H^-1(?) = Z is agreed.This theorem can be found in [30]. But professor Hu Yi go further. In [56], he prove the theorem does not only hold for hyperplane arrangement, but also holds for the subspace arrangement. Furthermore, in the same paper, he prove the theorem holds for more general condition-the mixed arrangement, which is adding spheres to the subspace arrangement. This is the first time that the concept of mixed arrangement appeared. Of course, Hu Yi is the first one who suggest the concept.This thesis is about a mathematics object, the mixed arrangement. The thesis consists of four chapters. In the first chapter, we'll introduce the hyperplane arrangement and subspace arrangement, which the mixed arrangement extend from. In the second chapter, we'll mainly research the properties of the mixed arrangement in combinatorics aspects. The third chapter contributes several results of the mixed arrangement in algebra aspects. Finally, some interesting open problems and several conjectures will be given in the fourth chapter. The concrete details follows:In the first chapter, we will review the history of the research of hyperplane arrangement and subspace arrangement. We'll introduce some problems, directions of the research and several famous results of the hyperplane arrangement and subspace arrangement on combinatorics, algebra and topology in the past. In the meantime, we also mention some work which was done recently in theses aspects.From the second chapter, we begin to consider the mathematic object, called the mixed arrangement, which is composed of the hyperplanes (or subspaces) and the spheres. We study the poset of its intersection set and compute the Mobius function of the mixed arrangement by the hyperplane (or subspace) arrangement's M(?)bius function. Moreover, by the method of deletion and restriction, we derive the recurse formula of the triple of the mixed arrangement and give the relationship between the characteristic polynomial and the number of the regions of the mixed arrangement.In the third chapter, we'll study the algebraic properties of the mixed ar-rangement. We define the derivation operator on the defining polynomial of the mixed arrangement. A special kind of mixed arrangement, which is called the free mixed arrangement, is our mainly study object. In this chapter, we prove that the Saito's criterion holds on the mixed arrangement just like which is correct to the hyperplane arrangement. And we study the relationship of module of derivation the among the triple (M, M', M"), when all of the three are free arrangement.Finally, in the fourth chapter, we propose several open problems which haven't been solved as our future work. As the extensive of the concept of the reflection hyperplane arrangement, we introduce the reflection mixed arrangement and give a conjecture about reflection mixed arrangement.