有限群和某些代数的中心

发布时间：2018-09-01 19:10

【摘要】：在代数学研究中有一个基本思想是通过比较,例如映射,同态和同构等,从一个熟知对象的性质和结构探究另一个对象的性质和结构等信息,以便更有效地将所研究的对象简单化,能进行有意义的分类,更为重要的实际意义是运用比较,对那些将要考察对象的性质与结构等有用信息的获取,可以找到或选取合适的对象(性质与结构等相对清楚的对象),进而从这个合适对象获取想要的信息。在我们研究群表示论中的体现是非常深刻的,当然在研究群代数及其他代数中也尤为重要。在有限维的情况下,通过对有限群表示理论的研究,分析其性质结构,给出群代数,群代数表示,群代数中心及Brauer代数和其中心维数等有关结论,主要目的是给出很详细的证明。具体安排如下:在第一章,我们首先介绍了本文相关的一些历史背景以及研究现状,然后简单的讲述了一下本文所做的工作。在第二章,我们给出了群表示的定义,并在此基础上给出子表示,商表示,表示的直和及表示的同态等定理,并给出相关证明。接着介绍群代数,给出群代数表示的定义,比较群表示与群代数表示的关系,在本章最后给出群代数中心的结构性质等相关证明。在第三章,我们给出Brauer代数的简单定义,及引用一些相关结果。接着我们又给出Brauer代数tB)(n及其中心的定义,随后研究t取任意参数时Brauer代数的中心维数,特别地,本章最后得出当t取特殊值时Brauer代数中心的维数大于或等于取普通值时的维数这一结论。在最后一章,我们对本文探讨的一些结果进行总结,并对今后可能研究的问题作进一步展望。
[Abstract]:In the study of algebra, a basic idea is to explore the properties and structures of one object from the nature and structure of a familiar object through comparison, such as mapping, homomorphism, and isomorphism. In order to simplify the objects studied more effectively, to be able to carry out meaningful classification, and the more important practical significance is to use comparison to obtain useful information such as the nature and structure of the objects to be examined, You can find or select the right object (relatively clear objects, such as nature and structure), and then get the desired information from the appropriate object. It is very profound in our study of group representation theory, but it is also very important in the study of group algebra and other algebras. In the case of finite dimension, through the study of the theory of finite group representation, the properties and structure of finite group representation theory are analyzed, and some conclusions such as group algebra, group algebra representation, center of group algebra, Brauer algebra and central dimension of group algebra are given. The main purpose is to give very detailed proof. The specific arrangements are as follows: in the first chapter, we first introduce some historical background and research status of this paper, and then briefly describe the work done in this paper. In chapter 2, we give the definition of group representation, and on this basis, we give the theorems of subrepresentation, quotient representation, direct sum of representations and homomorphism of representations, and give some relevant proofs. Then we introduce group algebra, give the definition of group algebra representation, compare the relations between group algebra representation and group algebra representation, and at the end of this chapter, we give some relevant proofs such as the structural properties of group algebra center. In chapter 3, we give a simple definition of Brauer algebra and cite some related results. Then we give the definition of Brauer algebra tB) (n and its center. Then we study the central dimension of Brauer algebra when t takes arbitrary parameters. Finally, the conclusion is drawn that the dimension of the center of Brauer algebra is greater than or equal to that of ordinary value when t takes a special value. In the last chapter, we summarize some results discussed in this paper, and make further prospects for the possible problems in the future.
【学位授予单位】：合肥工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.1

【相似文献】