EPSRC Reference: 
EP/W022834/1 
Title: 
KacMoody quantum symmetric pairs, KLR algebras and generalized SchurWeyl duality 
Principal Investigator: 
Przezdziecki, Dr T 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematics 
Organisation: 
University of Edinburgh 
Scheme: 
EPSRC Fellowship 
Starts: 
01 January 2023 
Ends: 
31 December 2025 
Value (£): 
299,742

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The idea of symmetry is one of the oldest and most fundamental ones in mathematics. It has its origin in geometry; for example, a square has eight symmetries  four reflections and four rotations. Symmetries have extra structure: they can be composed, and after applying a symmetry one can always reach the original state via an inverse symmetry. These properties are axiomatized in the algebraic concept of a group. In our example, the symmetries of a square give rise to a dihedral group. The process we have described can also be reversed  given a group or another algebraic object, we can realize it more concretely as a collection of symmetries. Such a realization is called a representation.
At the beginning of the twentieth century, Issai Schur and Hermann Weyl realized that there is a connection between the representations of two very important groups: the group of permutations of a collection of objects (the symmetric group) and the group of invertible matrices (the general linear group). Even though these groups are quite different, their representations are essentially the same. This relationship is now known as SchurWeyl duality, and constitutes one of the most persistent themes in representation theory, with countless generalizations in many different directions.
This project is concerned with one such generalization, whose origins are in statistical mechanics and quantum field theory. The sixvertex model describes the hydrogenbond configurations in a twodimensional sample of ice. The algebraic structure behind solutions to this model is the famous YangBaxter equation, which is, essentially, a representation of a braid group. It turns out that this representation is compatible with a representation of another object called a quantum group. If we enrich the sixvertex model by adding a boundary condition, the YangBaxter equation is replaced by the reflection equation, and the quantum group has to be upgraded to a quantum symmetric pair, i.e., a pair consisting of a quantum group and its coideal subalgebra.
The last decade has seen an explosion of interest in this area, as it became clear that most structures familiar from quantum group theory admit a generalization to quantum symmetric pairs. The goal of this project is to study the representation theory of quantum symmetric pairs in the context of SchurWeyl duality, using a variety of algebraic and geometric techniques. Another important component of our approach is categorification  a method which seeks to replace vector spaces by more universal structures like categories and functors. That is why KhovanovLaudaRouquier algebras, a fundamental tool in categorification, play a central role in the project.

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Organisation Website: 
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