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。群本身可以用生成器和关系来描述——即所谓的“群的表示”。给定一个群，人们可能想要理解它的“表示法”，即如何将群表示为复数域上的一组矩阵。拟议的研究将集中在这两个方面的定量问题：如何科学地呈现一个群体？需要多少个生成器和关系，它们的长度是多少？这些问题的动机来自计算群理论：高效的表示对高效的计算很重要。表征理论的定量方面处理的问题是：作为维度的函数，表征数量的增长率是多少？它是对处理置换表示的子群增长理论的阐述。通常在他们的研究中，陈述和陈述是非常脱节的。拟议研究的一个有趣的特点是两者之间的联系。如果成功，将对自由群的自同构群和映射类群的表示理论有新的启示。