Categorical and Diagrammatic Representation Theory

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Representation theory is the study of symmetries. Symmetry groups arise frequently in physics (e.g. rotations of a sphere), chemistry (e.g. crystallography), and other scientific fields. Data related to the objects possessing symmetry can often be encoded in an object called a representation. Mathematicians study the relationships between representations, and how bigger representations can be built from simple, indivisible ones, much as a molecule is built from indivisible atoms. Many properties of these simple representations, such as their dimensions, are unknown and the topic of intense research. The representations and their structure can be packaged in a collection called a category. An extremely fruitful tool of the last half century has been to identify categories in representation theory with categories from algebraic geometry, allowing the use of powerful geometric tools. But geometry also has its limits, especially when it comes to matters of explicit computation. In past work, the PI has found new and explicit descriptions of categories from representation theory and geometry, using diagrammatic methods. In diagrammatics, a very large matrix or a structure from geometry could be encoded as a picture and manipulated graphically. These descriptions make once-difficult categories accessible computer algebra systems. Computer calculations performed by the PI's collaborator Williamson have led to the first breakthroughs in computing dimensions of simple representations in decades. The PI will continue to develop diagrammatic methods to study representation theory and geometry, providing explicit constructions of new categories, structures, and tools which are beyond the current scope of other approaches. This project provides research training opportunities for students.More concretely, this proposal will support four related projects. The first is to provide general tools for studying generically semisimple monoidal categories diagrammatically using a cellular basis called the branching basis, akin to several bases previously constructed by the PI and collaborators. These tools will then be applied to the categories of singular Soergel bimodules, representations of symplectic groups, and representations of McKay groups. The second project is to introduce K-theoretic Soergel bimodules and to study their relationship to the quantum geometric Satake equivalence. The third is to produce a generalization of Khovanov-Lauda-Rouquier algebras, which has the potential to categorify other Nichols algebras. The fourth is to study the actions of lie algebras on various important categories, which were previously constructed by the PI and Qi.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.