Graphical and Categorical Methods in Representation Theory

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This project is in the general area of representation theory, which in broad terms is the idea of understanding symmetries of some naturally occurring structure such as a crystal by studying the different ways in which those symmetries can be realized in terms of matrices. By looking for the common features shared by all such matrix realizations at once, mathematicians have discovered some higher structures that encode all possible symmetries between symmetries. These new structures, "monoidal categories," can often be realized in purely graphical terms. In the last few years, this point of view has revealed some quite unexpected connections between very different parts of mathematics and physics. One example that plays a role in this project is the so-called Heisenberg category, which arose originally in algebra from thinking about symmetries between representations of the finite symmetric groups, but which is also connected to the infinite-dimensional Heisenberg Lie algebra, which has its origins in quantum mechanics. The funding of this project will lead to new understanding of classical problems in representation theory by taking advantage of this graphical approach, with potential applications both inside and outside of mathematics, including to finite group theory, combinatorics, Lie theory, knot theory, and theoretical physics. This project provides research training opportunities for graduate students.In more detail, the project will study the representation theory of algebraic groups, quantum groups and Lie superalgebras by exploiting some remarkable new pivotal monoidal categories which are defined by generators and relations. These include various Heisenberg categories whose definition is of a graphical nature. These monoidal categories act on many of the classically important categories in representation theory, leading to a unified general framework which reveals unexpected connections between different theories. The project will focus on five specific projects: Okounkov-Vershik approach to representations of the partition category; quasi-hereditary coalgebras and blocks of Deligne categories; the rational web category and thickened Heisenberg categorification; odd Heisenberg and super Kac-Moody 2-categories; Modular shifted Yangians and flag algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.