Categorical Diagonalization, Representation Theory, and Link Homology

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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.