CAREER: Categorical Representation Theory of Hecke Algebras

职业：赫克代数的分类表示论

• 批准号: 1553032
• 负责人: Benjamin Elias
• 金额: $ 46.29万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-04-01 至 2023-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1553032&HistoricalAwards=false
• 关键词: CAREER; Categorical; Representation; Theory; Hecke

Representation theory is, roughly speaking, the study of groups of symmetries. For example, a mirror on a wall can help to visualize a reflection of 3-dimensional space, a symmetry which sends each point in our world to the corresponding point on the other side of the mirror. Coxeter groups are special kinds of groups of symmetries that consist of sequences of reflections through mirrors placed at precise angles to one another. The so-called "crystallographic" Coxeter groups occur frequently in physics and mathematics because they preserve lattice points, for example the locations of atoms in a crystal. Crystallographic Coxeter groups also arise when analyzing important geometric spaces; this imbues them with a great deal of interesting structure. Mysteriously, Coxeter groups which are not crystallographic still possess this beautiful structure, despite having no geometric explanation. For example, each Coxeter group has an associated Hecke algebra, and by multiplying elements in this algebra one can produce numbers called structure coefficients. For crystallographic Coxeter groups, these coefficients are always non-negative because they are counting something. However, the non-negativity result holds in general. An underlying goal of this project is to find combinatorial and algebraic descriptions of the structures apparent in Coxeter groups, to help explain these mysterious phenomena. The main objects of study are categorical representations of Hecke algebras, where usual reflections are replaced by "reflection functors" that act as symmetries not on some n-dimensional space, but on spaces attached to representations of other important algebras in mathematics.In this research project the categorical representation theory of Hecke algebras will be investigated along three lines of attack. The first approach is to lift the notion of diagonalization from linear algebra to categorical representation theory. For example, the full twist in the braid group, and its image in the Hecke algebra, is diagonalizable in any finite dimensional representation. In joint work with Hogancamp, I aim to prove that the categorified full twist is "categorically diagonalizable." This allows one to lift much of the structure theory of Hecke algebra representations to the categorical level. In the second approach, together with Williamson and Juteau, I will study certain categorical representations of affine Weyl groups which are significant for modular representation theory, and their quantum deformations. The goal is to compute local intersection forms, which will explain how the category degenerates in finite characteristic. This computation will give character formulas for modular representations of algebraic groups which were previously unknown. In the third approach, together with Young, I will find an algebraic description of the quantum deformation mentioned above, when the quantum parameter is a root of unity. This is related to the categorification of complex reflection groups, which is an open problem.

大概是对对称组的研究。例如，墙上的镜子可以帮助您形象地看到3维空间的反射，这是一种将我们世界中的每个点传递到镜子另一侧的相应点。 Coxeter组是特殊类型的对称组组，包括通过彼此精确的镜子组成的反射序列。所谓的“晶体学” Coxeter组经常出现在物理和数学中，因为它们保留了晶格点，例如原子在晶体中的位置。在分析重要的几何空间时，也会出现晶体学交换组。这使他们充满了许多有趣的结构。神秘的是，尽管没有几何解释，但不是晶体学的高级群仍然具有这种美丽的结构。例如，每个Coxeter组都有相关的Hecke代数，并且通过将该代数中的元素乘以产生称为结构系数的数字。对于晶体学Coxeter组，这些系数始终是非负的，因为它们正在计算某些内容。但是，通常会产生非阴性结果。该项目的一个根本目标是找到对Coxeter组明显的结构的组合和代数描述，以帮助解释这些神秘的现象。研究的主要对象是Hecke代数的分类表示，在该研究中，通常的反思被“反射函数”代替，而不是在某些n维空间上，而是在数学中其他重要代数的代表的空间上。项目将沿着三条攻击线进行分类代数的分类代表理论。第一种方法是将对角度化的概念从线性代数到分类代表理论。例如，编织组的完整扭曲及其在Hecke代数中的图像在任何有限的尺寸表示中都是对角线的。在与Hogancamp的联合合作中，我的目标是证明该分类的完全扭曲是“可对角度可对角的”。这使人们可以将Hecke代数表示的许多结构理论提升到分类级别。 在第二种方法中，我将与威廉姆森（Williamson）和柔托（Juteau）一起研究仿生韦尔基团的某些分类表示，这些表示对于模块化表示理论及其量子变形很重要。目标是计算局部交叉表格，这将解释类别如何在有限特征中退化。该计算将为代数基团的模块化表示为特征公式，而代数组的模块化表示。在第三种方法中，与Young一起，我将发现上述量子变形的代数描述，当时量子参数是统一的根。这与复杂反射组的分类有关，这是一个空旷的问题。