Symmetry Of Array Space And Computation Of Related Invariants

The theory of arrays has a significant future in applications. Especially in recent years, with the development of computer science, it plays more and more important roles in data classification, information theory and coding theory. But few research on the property of arrays has been done. This thesis tries to study the symmetry of array space by using group action on array spaces.The dihedral group, which consists of rotations and reflections, can describe some geometric symmetry. Considering the action of dihedral group on some concret array spaces, this paper studies the symmetry of array space, computes the cycle index and points out some applications of cycle index. First, this paper considers the action of dihedral group on set of indexes of array, computes the cycle index respectively, and then considers mappings from a set of index to the set {0,1}. These mappings compose a special set of arrays. Because each mapping can associate with a subgraph of complete digraphs or undirected graphs, we compute the equivalant class of graphs using cycle index theorem. Furthermore, this paper also studies a class of array spaces and computes the cycle index of dihedral group which acts on the array space. This paper computes the cardinal of the cohomology group H~0, which has a strong significance for studying the symmetry of array spaces.The study of the symmetry of array space under group actions in this thesis provides a new method for studying array space.