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.

表示论，粗略地说，是研究群的对称性。例如，墙上的一面镜子可以帮助想象三维空间的反射，这种对称性将我们世界中的每个点发送到镜子另一边的相应点。考克斯特群是特殊类型的对称群，由通过彼此成精确角度放置的镜子的反射序列组成。所谓的“晶体学”考克斯特群经常出现在物理学和数学中，因为它们保留了晶格点，例如晶体中原子的位置。在分析重要的几何空间时，也会出现晶体学的考克斯特群;这使它们具有许多有趣的结构。令人惊讶的是，非晶体学的考克斯特群仍然拥有这种美丽的结构，尽管没有几何解释。例如，每个考克斯特群都有一个相关的赫克代数，通过乘以这个代数中的元素，可以产生称为结构系数的数字。对于晶体学的考克斯特群，这些系数总是非负的，因为它们在计数。然而，非负性的结果在一般情况下成立。这个项目的一个潜在目标是找到考克斯特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。主要的研究对象是Hecke代数的范畴表示，其中通常的反射被“反射函子”所取代，这些反射函子不是在某些n维空间上，而是在数学中其他重要代数的表示所连接的空间上作为对称。在这个研究项目中，Hecke代数的范畴表示理论将沿着沿着三条攻击线进行研究。第一种方法是将对角化的概念从线性代数提升到范畴表示论。例如，辫子群中的全扭曲，以及它在Hecke代数中的图像，在任何有限维表示中都可对角化。在与Hogancamp的合作中，我的目标是证明分类的全扭转是“绝对可对角化的”。“这使得人们可以将Hecke代数表示的大部分结构理论提升到范畴水平。 在第二种方法中，我将与威廉姆森和朱托一起研究仿射外尔群的某些范畴表示及其量子变形，这些范畴表示对模表示理论具有重要意义。目标是计算局部交形式，这将解释范畴如何在有限特征下退化。这种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Young一起找到上述量子形变的代数描述。这与复杂反射群的分类有关，这是一个开放的问题。