CAREER: Categorical Representation Theory of Hecke Algebras

粗略地说，表象理论是对对称群的研究。例如，墙上的镜子可以帮助可视化三维空间的反射，这种对称性将我们世界中的每个点都发送到镜子另一边的相应点。Coxeter群是一种特殊类型的对称组，它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“结晶学”Coxeter基团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，结晶学Coxeter基团也会出现；这给它们注入了大量有趣的结构。神秘的是，尽管没有几何解释，但非结晶学的Coxeter基团仍然拥有这种美丽的结构。例如，每个Coxeter群都有一个相关的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。对于结晶学Coxeter基团，这些系数总是非负的，因为它们在计数。然而，非负性结果大体上是成立的。这个项目的一个基本目标是找到Coxeter群中明显结构的组合和代数描述，以帮助解释这些神秘的现象。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”取代，这些反射函子不是在某个n维空间上而是在数学中其他重要代数的表示所依附的空间上充当对称性。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理论。例如，辫子群中的全捻及其在Hecke代数中的像在任何有限维表示中都是对角化的。在与Hogancamp的合作中，我的目标是证明范畴化的全扭曲是“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层面。在第二种方法中，我将与Williamson和Juteau一起研究仿射Weyl群的某些范畴表示，这些表示对于模表示理论及其量子形变具有重要意义。目标是计算局部交形式，这将解释范畴如何在有限特征中退化。这一计算将给出以前未知的代数群的模表示的特征标公式。在第三种方法中，我将与Young一起找到上述量子形变的代数描述，当量子参数是单位根时。这与复杂反射群的分类有关，这是一个悬而未决的问题。

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.