CAREER: Categorical Representation Theory of Hecke Algebras

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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.