Categorical Diagonalization, Representation Theory, and Link Homology

表征理论的主题构成了讨论对称性的数学基础。例如，一个正方形有八种对称，由“旋转90度”或“沿对角线反射”等变换组成，以及这些变换的组合。这些对称中的每一个都代表了一个平面的变换使正方形保持不变，我们说平面的变换形成了正方形对称群的表示。正方形和其他多边形的对称性是一类被称为Coxeter群的群的特殊例子，它们捕获并推广了反射群的直观概念。近几十年来，数学家们发现了一个丰富的表征理论，在这个理论中，被作用的对象不是一个平面（或一些高维的类似物），而是一种更有结构的对象，称为范畴。这个项目关注的是Coxeter群的范畴表示理论和一些密切相关的对象，称为Hecke代数，以及与其他数学领域的联系，例如拓扑中的结和连接的研究。更详细地说，将研究三个相互关联的对象：(a) Soergel双模的范畴，(b)平面上点的Hilbert格式，以及(c) Khovanov-Rozansky链路同调。首先，研究者将继续发展范畴对角化理论，并将结果应用于Hecke代数和量子群的范畴表示理论。这包括对分类卡西米尔算子的研究。作为范畴对角化的一个应用，作用于Soergel双模范畴的全扭转Rouquier复合体将被对角化，扩展已经在a型中完成的工作。由此产生的特征分解提供了一种接近Gorsky， Negut和Rasmussen最近关于Soergel双模与Hilbert格式之间深度对应的猜想的方法。研究者将利用分类对角化，以及最近的计算突破，努力证明这种对应关系。最后，Gorsky-Negut-Rasmussen对应对三阶Khovanov-Rozansky同调的结构做出了一些预测，研究者将利用与Hilbert方案的联系来探索这些预测。

The subject of representation theory forms the mathematical basis for discussing symmetry. As an example, there are eight symmetries of a square, consisting of transformations like "rotate by 90 degrees," or "reflect across a diagonal line," and combinations of these. Each of these symmetries represents a transformation of the plane that leaves the square unchanged, and we say that the transformations of the plane form a representation of the symmetry group of the square. The symmetries of the square and other polygons are special examples of a family of groups called Coxeter groups, which capture and generalize the intuitive notion of a reflection group. In recent decades, mathematicians have discovered a rich theory of representations in which the object being acted on is not a plane (or some higher dimensional analogue), but rather a more structured sort of object, called a category. This project is concerned with the categorical representation theory of Coxeter groups and some closely related objects, called Hecke algebras, and connections to other areas of mathematics, such as the study of knots and links in topology.In more detail, three interrelated objects will be studied: (a) categories of Soergel bimodules, (b) Hilbert schemes of points in the plane, and (c) Khovanov-Rozansky link homology. First, the investogator will continue to develop the theory of categorical diagonalization and apply the results to the categorified representation theory of Hecke algebras and quantum groups. This includes work on the categorified Casimir operator. As an application of categorical diagonalization, the full-twist Rouquier complex acting on categories of Soergel bimodules will be diagonalized, extending work already accomplished in type A. The resulting eigendecompositions present a method for approaching recent conjectures of Gorsky, Negut, and Rasmussen regarding a deep correspondence between Soergel bimodules and Hilbert schemes. The investigator will utilize categorical diagonalization, as well as recent computational breakthroughs, to work toward a proof of this correspondence. Finally, the Gorsky-Negut-Rasmussen correspondence makes several predictions regarding the structure of the triply graded Khovanov-Rozansky homology, which the investigator will explore using insights from the connection with Hilbert schemes.