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’.
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.
