Groups: representations and presentations

团体：陈述和演示

• 批准号: 0801190
• 负责人: Gregory Margulis
• 金额: $ 17.94万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801190&HistoricalAwards=false
• 关键词: Groups; representations; presentations

Groups are basic objects in mathematics - expressing symmetries ofvarious objects. The groups themselves can be described by generatorsand relations - so called "presentations of groups". Given a group, onemay want to understand its "representation" i.e. the way to presentthe group as a group of matrices over the field of complex numbers.The proposed research will focus on quantitative questions of bothaspects: How e±ciently can a group be presented? How many genera-tors and relations are needed and what is their length? The motivationfor these questions comes from computational group theory: an efficientpresentation is important for efficient computation.The quantitative aspects of representation theory deal with questionsof the form: What is the rate of growth of the number of representationsas a function of the dimension? It is an elaboration of the theory ofsubgroup growth which deals with permutational representations.Usually presentations and representations are very much disjoint intheir studies. One of the interesting features of the proposed researchis a connection between the two. If successful, it can shed new light onthe representation theory of the automorphism group of the free groupand of the mapping class groups.

群是数学中的基本对象--表示各种对象的对称性。群本身可以用生成元和关系来描述，即所谓的“群的表示”。给定一个群，人们可能想理解它的“表示”，即把群表示为复数域上的矩阵群的方法，本文的研究将集中在这两个方面的定量问题上：如何精确地表示一个群？需要多少个生成元和关系，它们的长度是多少？这些问题的动机来自计算群论：一个有效的表示对于有效的计算是很重要的。表示论的定量方面处理的问题的形式：什么是增长率的代表作为一个函数的维数？它是子群增长理论的一个阐述，涉及置换表示。通常表示和表示在他们的研究中是非常不相交的。这项研究的一个有趣的特点是两者之间的联系。如果成功的话，它将为自由群的自同构群和映射类群的表示理论提供新的线索。