Categorical Diagonalization, Representation Theory, and Link Homology

表示论的主题形成了讨论对称性的数学基础。 例如，一个正方形有八种对称，包括“旋转90度”或“沿对角线反射”等变换，以及这些变换的组合。 这些对称中的每一个都代表了平面的一个变换，使正方形保持不变，我们说平面的变换形成了正方形的对称群的一个表示。正方形和其他多边形的对称性是称为考克斯特群的群族的特例，考克斯特群捕捉并推广了反射群的直观概念。近几十年来，数学家们发现了一个丰富的表示理论，在这个理论中，作用于其上的对象不是一个平面（或某种更高维的类似物），而是一种更有结构的对象，称为范畴。 这个项目涉及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，例如拓扑学中的结和链接的研究。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方案，和（c）Khovanov-Rozansky链接同调。 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。这包括分类的Casimir算子的工作。 作为范畴对角化的一个应用，我们将对角化作用在Soergel双模范畴上的全扭Rouquier复形，扩展了在类型A中已经完成的工作。 由此产生的本征分解提出了一种方法，用于接近最近的高斯基，内古特，和拉斯穆森关于Soergel双模和希尔伯特计划之间的深层对应关系。研究人员将利用分类对角化以及最近的计算突破来证明这种对应关系。 最后，Gorsky-Negut-Rasmussen对应对三重分次Khovanov-Rozansky同调的结构做出了几个预测，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.