Diagram algebras via Soergel bimodules

Dr Amit Hazi
Research Fellow 2019

Dr Amit Hazi
City University, London

The concept of symmetry is omnipresent in the sciences. The mathematical study of symmetry is called representation theory. Representation theory seeks to better understand abstract algebraic objects by representing them concretely as collections of symmetries of more familiar objects. Besides its intrinsic utility within mathematics itself, representation theory also underpins our current understanding of physics and has applications in chemistry and crystallography.

Representation theory over the complex numbers is well understood for the most fundamental algebraic objects. For example, the earliest construction of the representations today known as standard modules dates back over a century. Over the complex numbers these are irreducible, forming the basic building blocks for all representations. However, over other number systems exhibiting ‘clock’ arithmetic, the standard modules are no longer irreducible. Determining how they decompose into irreducible representations is the most important unsolved problem in representation theory today. The most successful approaches to this problem exploit ‘higher’ symmetry in representation theory. More precisely, we can view the standard modules as points in a certain vector space called weight-space. Reflection symmetry in weight-space is a type of symmetry of the representations, or in other words ‘symmetry of symmetries’.

 

“The most successful approaches to this problem exploit ‘higher’ symmetry in representation theory”

For several decades Lusztig’s conjecture suggested a solution to the decomposition problem in terms of this higher symmetry. However, in an astounding recent development Williamson showed that Lusztig’s conjecture is false. The best way to fix Lusztig’s conjecture and solve the decomposition problem is to better understand when analogues of Lusztig’s conjecture do hold. This occurs for many diagram algebras, which are constructed from diagrams involving ‘strings’. More recently, Soergel, Williamson, and others gave a method for building diagrammatic structures called Soergel bimodules for which Lusztig’s conjecture always holds. They have also shown that Soergel bimodules are intricately related to the decomposition problem. We propose to search for direct correspondences between diagram algebras and Soergel bimodules, and to apply these correspondences to the decomposition problem. Several such correspondences have already been conjectured, but few have actually been established.

This ambitious project will greatly advance representation theory by unifying classical diagram algebras under the modern machinery of Soergel bimodules.