Collaborative Research: Rank and Duality in Representation Theory

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（线、圆、正方形）都是对称的。人们很少意识到对称是理解世界的基础。相对论和量子力学是20世纪物理学的两大发展，它们在很大程度上都依赖于对称的思想。线性代数是19世纪末和20世纪的另一个数学发展，现在在整个科学中被大量使用。表征理论研究的是如何将对称与线性代数结合起来。这个项目处理不同物体的对称系统之间的对应关系，称为eta对应关系。这个项目的第一阶段是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。项目的后续阶段将扩展eta对应到更多的案例，改进用于描述它们的概念，并使用它们将表示理论应用于纯数学和应用数学中的广泛问题。更详细地说，这个项目引入了一种创新的方法来研究有限域和局部域上经典群的表示，这种方法似乎有利于谐波分析。一个有效的“大小”表示理论将被开发，包括一个精确的定义和方法来分析给定大小的表示。有限设置的动机来自于这样一个事实，即关于有限群的许多问题（例如，随机漫步，单词映射，Cayley图等）可以使用调和分析来处理。更准确地说，干预这些问题的是相关组g的不可约表示（reps）的字符比率（字符除以维数）。一般来说，精确地计算字符比率是不可行的，但对于应用程序来说，通常足以表明大多数表示的字符比率很小。由于在许多情况下，表示的维度是使字符比例变小的原因，因此第一阶段是了解reps的维度，特别是那些尺寸比平均值小得多的尺寸，因为它们最有可能对任何字符比例总和做出主要贡献。研究人员的理论适用于所有经典群，甚至可能适用于有限域和局部域上的所有约化群。他们提出了几种不同的表示的秩的概念，他们怀疑，虽然在性质上不同，这些概念是等价的。有了这些概念，就提供了关于G的不可逆环维数的大量信息。此外，研究人员发现了一个系统结构，称为eta对应，在给定秩的G的自然定义的大族不可逆环和（全部或大部分）较小群h的不可逆环之间。有理由相信这种结构是详尽的，该项目追求对这一猜想的证明。eta对应关系为它所构建的表示提供了对字符比率的强大控制，对这种关系的正式处理将形成项目的第二阶段。到目前为止，一个重要的发现是，尽管给定等级的不原状的尺寸差异很大，但这些不原状的字符比率几乎是相等的。因此，为了调和分析的目的，固定秩的表示形成一个自然的族来研究。最后，在项目的第三阶段，研究人员将把特征比率和维度的界限应用于群论及其应用中的几个开放问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.