Representations of Iwahori-Hecke algebras

Representation Theory is a fruitful and interesting field of pure mathematics which has potential applications to various other fields such as quantum chemistry, physics, and number theory. The representation theory of the symmetric groups is a particularly rich area of research which utilises the symmetries of partitions, and connects the broader areas of algebra and combinatorics together.
One specific way to study the (irreducible) representations of the symmetric groups S_n, for some positive integer n, is by looking at (simple) modules over the algebra KS_n for some field K. In the case where K is an algebraically closed field of characteristic greater than n, also known as the ordinary representations, irreducible representations of S_n are very well understood, including their degrees, character formulae, and more. However, the case where the characteristic of K is less than or equal to n, known also as the modular representations, is not as well understood. For example, the dimensions of irreducible modular representations are not known in general.
One particular example is the family of Specht modules. These naturally-defined modules turn out to be all of the simple modules for the ordinary representations of symmetric groups. However, in the modular case, these Specht modules may no longer be irreducible, so studying the decomposition of Specht modules into irreducibles becomes a very interesting question. These are called "decomposition numbers" and studying these is a key goal of this research project.
The representation theory of Iwahori-Hecke algebras of the symmetric groups includes the modular representation theory of S_n as a special case. By studying the representations of a broader collection of algebras, we gain more insight into the particular case of S_n.
The first part of this research project closely follows the book "Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group" by Andrew Mathas. In particular, we focus on chapters 1, 2, 3, and 6.
The first chapter studies the symmetric group as an abstract group, reminding the reader of various properties that follow from the braid relations on the generators of the group. The author then constructs the Iwahori-Hecke algebra, using definitions analogous to those of the symmetric group.
The second chapter studies a family of algebras called "cellular algebras". A cellular algebra A is an algebra together with a distinguished cellular basis, which is well adapted to studying the representation theory of A. The Iwahori-Hecke algebras are an example of cellular algebras. There are two important features of cellular algebras. Firstly, the cellular basis which gives a filtration of A with composition factors isomorphic to the cell modules of A. Secondly, we can define bilinear forms on each of the cell modules, which allow us to construct all of the simple A-modules.
The third chapter looks at the modular representation theory of the Iwahori-Hecke algebra. We study these by looking at the combinatorics of tableaux, Specht modules and Jucys-Murphy elements, and irreducible modules over the Iwahori-Hecke algebra.
Finally, the sixth chapter focuses on branching rules, canonical bases and decomposition matrices. The final section of chapter six looks at Ariki-Koike algebras, a further deformation of particular types of Iwahori-Hecke algebras. This is a potential avenue for further research.
In summary, the goal of my research project is to understand the (modular) representation theory of the symmetric groups. We do this by studying modules over a broader collection of algebras, namely the Iwahori-Hecke algebra, which has nice structure due to being a cellular algebra. We use a variety of techniques, such as the combinatorics of tableaux, and look at interaction between ordinary representations and modular representations through decomposition numbers. There is also potential to continue research in this direction by looking at Ariki-Koike algebras.