Collaborative Research: Rank and Duality in Representation Theory

合作研究：表示论中的等级和对偶性

• 批准号: 1805004
• 负责人: Roger Howe
• 金额: $ 4.87万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1805004&HistoricalAwards=false
• 关键词: Collaborative; Research; Rank; Duality; Representation

This project concerns representation theory. Symmetry is a familiar concept in elementary geometry -- many of the most important figures (lines, circles, squares) are symmetrical. It is less widely appreciated that symmetry has been found to be fundamental for understanding the world. Both the theory of relativity and quantum mechanics, two major developments in physics during the 20th century, rely heavily on ideas of symmetry. Linear algebra is another mathematical development of the late 19th and 20th centuries that is now heavily used throughout science. Representation theory is the study of how symmetry can be combined with linear algebra. This project deals with correspondences, called eta correspondences, between systems of symmetries of different objects. The first phase of this project is to show that there are eta correspondences for some of the most important finite systems of symmetries, and how to describe these correspondences. Subsequent phases of the project will extend eta correspondences to more cases, refine the concepts used to describe them, and use them to apply representation theory to a broad range of questions in pure and applied mathematics.In more detail, this project introduces an innovative approach to the study of representations of classical groups over finite and local fields, an approach that seems beneficial for harmonic analysis. An effective theory of "size" for representations will be developed, including a precise definition and a method to analyze representations of a given size. The motivation in the finite setting comes from the fact that many questions about finite groups (e.g., random walks, word maps, Cayley graphs, etc.) can be approached using harmonic analysis. More precisely, what intervenes in such problems are the character ratios (character divided by dimension) of the irreducible representations (irreps) of the relevant group G. In general, it is not feasible to compute the character ratios exactly, but for applications it often suffices to show that the character ratios are small for most representations. Since in many cases the dimension of the representation is what makes the character ratio small, the first phase is to understand the dimensions of irreps and, especially, those with dimensions that are much smaller than average, since they most likely to make the dominant contributions to any sum of character ratios. The investigators have a theory that is applicable to all classical groups and, perhaps, even to all reductive groups over finite and local fields. They propose several different notions of rank of a representation, and they suspect that, although different in nature, these notions are equivalent. Having these notions in hand gives a lot of information on the dimensions of the irreps of G. In addition, the investigators discovered a systematic construction, called the eta correspondence, between large naturally defined families of irreps of G of a given rank, and (all, or most of) the irreps of a smaller group H. There is reason to believe that this construction is exhaustive, and the project pursues a proof of this conjecture. The eta correspondence gives strong control over character ratios for the representations it constructs, and a formal treatment of this relation will form the second phase of the project. A significant discovery so far is that although the dimensions of irreps of a given rank vary considerably, the character ratios of these irreps are nearly equal. Thus, for purposes of harmonic analysis, representations of a fixed rank form a natural family to study. Finally, in the third phase of the project, the investigators will apply bounds on character ratios and dimensions to several open problems in group theory and its applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目涉及表征理论。对称是初等几何中一个熟悉的概念--许多最重要的图形（直线、圆、正方形）都是对称的。对称性被认为是理解世界的基础，但这一点却没有得到广泛的认识。相对论和量子力学是20世纪世纪物理学的两大发展，它们都在很大程度上依赖于对称性的思想。线性代数是19世纪末和20世纪的另一个数学发展，现在在整个科学中大量使用。表示论是研究如何将对称性与线性代数结合起来。这个项目处理的对应，称为eta对应，不同对象的对称系统之间。这个项目的第一阶段是要证明一些最重要的有限对称系统存在eta对应，以及如何描述这些对应。该项目的后续阶段将扩展eta对应到更多的情况下，细化用于描述它们的概念，并使用它们来应用表示理论在纯数学和应用数学的广泛问题。更详细地说，该项目介绍了一种创新的方法来研究有限和局部域上的经典群的表示，这种方法似乎有利于调和分析。一个有效的理论的“大小”的表示将开发，包括一个精确的定义和一种方法来分析一个给定的大小表示。有限设置的动机来自于这样一个事实，即许多关于有限群的问题（例如，随机游走、单词映射、Cayley图等）可以使用谐波分析来处理。更确切地说，介入这些问题的是相关群G的不可约表示（不可约表示）的特征标比（特征标除以维数）。一般来说，精确计算字符比率是不可行的，但对于应用程序来说，对于大多数表示来说，字符比率通常很小。由于在许多情况下，表征的维度是使字符比率变小的原因，因此第一阶段是了解非表征的维度，特别是那些维度远小于平均值的维度，因为它们最有可能对任何字符比率的总和做出主要贡献。研究人员有一个理论，适用于所有的经典群，甚至可能适用于所有的还原群在有限和局部领域。他们提出了几个不同的概念等级的代表性，他们怀疑，虽然不同的性质，这些概念是等价的。掌握了这些概念，我们就能得到关于G的非代表维数的大量信息。此外，研究者们还发现了一个系统的构造，称为eta对应，它存在于给定秩的G的自然定义的大的非代表族和较小的群H的（全部或大部分）非代表族之间。有理由相信，这个结构是详尽的，该项目追求证明这一猜想。eta对应为它所构造的表示提供了对字符比率的强有力控制，对这种关系的正式处理将形成该项目的第二阶段。到目前为止，一个重要的发现是，尽管给定等级的irreps的尺寸变化很大，但这些irreps的特征比例几乎相等。因此，为了调和分析的目的，固定秩的表示形成了一个自然的族来研究。最后，在项目的第三阶段，研究人员将应用字符比例和尺寸的界限，在群论及其应用中的几个开放问题。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。