Categorical Diagonalization, Representation Theory, and Link Homology

表象理论是讨论对称性的数学基础。举个例子，正方形有八种对称，由“旋转90度”或“沿对角线反射”等变换以及它们的组合组成。这些对称中的每一个都表示保持正方形不变的平面的变换，我们说该平面的变换形成了正方形的对称群的表示。正方形和其他多边形的对称性是称为Coxeter群的群族的特殊示例，它捕捉并推广了反射群的直观概念。近几十年来，数学家们发现了一种丰富的表示理论，在这种理论中，被作用的对象不是平面(或某种更高维度的类比)，而是一种更有结构的对象，称为范畴。这个项目涉及Coxeter群和一些密切相关的对象的范畴表示理论，称为Hecke代数，以及与其他数学领域的联系，如拓扑学中的纽结和环的研究。更详细地，将研究三个相互关联的对象：(A)Soerel双模范畴，(B)平面上点的Hilbert方案，(C)Khovanov-Rozansky环同调。首先，研究者将继续发展范畴对角化理论，并将其结果应用于Hecke代数和量子群的范畴化表示理论。这包括关于范畴Casimir算子的工作。作为范畴对角化的一个应用，作用在Soerel双模范畴上的全扭Rouquier复形将被对角化，推广了A型中已经完成的工作。由此产生的特征分解提供了一种方法来逼近最近Gorsky，Negut和Rasmussen关于Soerel双模与Hilbert方案之间的深度对应的猜想。研究人员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对应对三级Khovanov-Rozansky同调的结构做出了几个预测，研究人员将利用与Hilbert方案的联系的见解来探索这些预测。

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.