Finite Groups And The Center Of Some Algebras

The basic idea is the way of comparison in the study of algebra,for example,mapping,homomorphism and isomorphism.We explore some characters and structures of another object from the nature and structure of the known object,in order to make the research object more easily and get a meaningful classification.Above all,by comparison,those who will be studied about the properties and structure of useful information can be found,if we select the appropriate object(the object properties and the structure is relatively clear).This thought is very profound in our study of group representation,of course,it is also important to study group algebra or other algebra.In the finite dimensional case by finite group representation theory,analysis of the nature of its structure,we give the group algebra,the representation of group algebra,Brauer algebra,the center of Brauer algebra and its dimension related conclusions,the main purpose is to give a detailed proof.The thesis is organized as follows:In Chapter 1,some related backgrounds and recently progresses are introduced,and we give a brief introduction about what we have done in this thesis.In Chapter 2,we give the definition of group representation,and on the basis of this,we give some theorems,such as child representation,quotient representation,direct sum of representation and homomorphism of representation.Then,we introduce the definition of group algebra,and compare the relation between group representation and group algebra representation.In the last chapter,we give the proof of the structures and properties of the center of group algebra.In Chapter 3,we introduce a simple definition of algebra and some related results.the definition of Brauer algebra and its center is again introduced,then we discuss the dimension of the center when the parameters is arbitrary,especially the dimension of the algebraic center is greater than or equal to the exact dimension of the normal value when the special parameter is taken in the last of this chapter.In the last Chapter,we give a summary for this paper and list several questions which to be solved.