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.

粗略地说，表示论是对对称群的研究。例如，墙上的镜子可以帮助可视化 3 维空间的反射，这种对称性将我们世界中的每个点发送到镜子另一侧的相应点。考克塞特群是一种特殊的对称群，由通过彼此精确角度放置的镜子的反射序列组成。所谓的“晶体学”考克塞特群经常出现在物理和数学中，因为它们保留晶格点，例如晶体中原子的位置。晶体学考克塞特群在分析重要的几何空间时也会出现；这给它们注入了很多有趣的结构。神秘的是，非晶体学的考克塞特群仍然拥有这种美丽的结构，尽管没有几何解释。例如，每个 Coxeter 群都有一个关联的 Hecke 代数，通过将该代数中的元素相乘，可以产生称为结构系数的数字。对于晶体学 Coxeter 群，这些系数总是非负的，因为它们正在计算某些东西。然而，非负结果总体成立。该项目的根本目标是找到 Coxeter 群中明显结构的组合和代数描述，以帮助解释这些神秘现象。研究的主要对象是赫克代数的分类表示，其中通常的反射被“反射函子”取代，“反射函子”不是在某个 n 维空间上充当对称性，而是在数学中其他重要代数的表示所附加的空间上起作用。在这个研究项目中，赫克代数的分类表示理论将沿着三个方向进行研究。第一种方法是将对角化的概念从线性代数提升到分类表示理论。例如，辫子群中的全扭曲及其在赫克代数中的图像在任何有限维表示中都是可对角化的。在与 Hogancamp 的合作中，我的目标是证明分类全扭曲是“绝对可对角化的”。这使得人们能够将赫克代数表示的大部分结构理论提升到分类水平。 在第二种方法中，我将与 Williamson 和 Juteau 一起研究仿射 Weyl 群的某些分类表示，这对于模表示理论及其量子变形具有重要意义。目标是计算局部交集形式，这将解释类别如何在有限特征中退化。该计算将给出以前未知的代数群的模表示的特征公式。在第三种方法中，当量子参数是单位根时，我将与杨一起找到上述量子变形的代数描述。这与复杂反射群的分类有关，这是一个悬而未决的问题。