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Details of Grant 

EPSRC Reference: EP/W022834/1
Title: Kac-Moody quantum symmetric pairs, KLR algebras and generalized Schur-Weyl duality
Principal Investigator: Przezdziecki, Dr T
Other Investigators:
Researcher Co-Investigators:
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:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
11 Apr 2022 EPSRC Mathematical Sciences Fellowship Interview Panel April 2022 Announced
08 Feb 2022 EPSRC Mathematical Sciences Prioritisation Panel February 2022 Announced
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 Schur-Weyl 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 six-vertex model describes the hydrogen-bond configurations in a two-dimensional sample of ice. The algebraic structure behind solutions to this model is the famous Yang-Baxter 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 six-vertex model by adding a boundary condition, the Yang-Baxter 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 Schur-Weyl 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 Khovanov-Lauda-Rouquier algebras, a fundamental tool in categorification, play a central role in the project.

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Organisation Website: http://www.ed.ac.uk