CAREER: Categorical Representation Theory of Hecke Algebras

粗略地说，表征理论是对对称群的研究。例如，墙上的一面镜子可以帮助想象三维空间的反射，这种对称将我们世界中的每个点都传送到镜子另一边的相应点上。考克斯特群是一种特殊的对称群，它由一系列反射镜以精确的角度相互放置而成。所谓的“晶体学”考克斯特群在物理学和数学中经常出现，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，也会出现晶体科塞特群；这给他们注入了大量有趣的结构。神秘的是，尽管没有几何解释，但非晶体学的考克斯特群仍然具有这种美丽的结构。例如，每个Coxeter群都有一个相关的Hecke代数，通过将这个代数中的元素相乘，可以得到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，因为它们在计算一些东西。然而，一般来说，非否定的结果是成立的。该项目的一个潜在目标是找到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维空间上起对称作用，而是在与数学中其他重要代数表示相连的空间上起对称作用。在这个研究项目中，赫克代数的范畴表示理论将沿着三条路线进行研究。第一种方法是将对角化的概念从线性代数提升到范畴表征理论。例如，辫群中的全捻及其在Hecke代数中的像在任何有限维表示中都是可对角的。在与Hogancamp的合作中，我的目标是证明分类的完全扭转是“绝对对角化的”。这使得人们可以将赫克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Williamson和Juteau一起研究仿射Weyl群的某些对模表示理论有重要意义的分类表示及其量子变形。目标是计算局部相交形式，这将解释在有限特征下范畴是如何退化的。这个计算将给出以前未知的代数群的模表示的字符公式。在第三种方法中，我将与Young一起找到上述量子变形的代数描述，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.