Representations of the symmetric group on n letters over the complex field are well-understood: for every partition of n, we define a module, known as a Specht module, and these Specht modules give a complete set of pairwise non-isomorphic irreducible modules. There exist closed formulae for their dimensions and methods to compute their characters. However, representations of the symmetric group over fields of positive characteristic are not well-understood. For example, even though it is possible to construct the irreducible modules explicitly as quotients of the Specht modules, their dimensions are not generally known. A constructive approach to this problem was given by James [i] who developed the use of combinatorial tools, such as diagrams, tableaux and abacuses. This approach generalises in a straightforward way to give techniques for studying representations of related algebras including the Hecke algebras of type A and the Ariki-Koike algebras. See the book [ii] and the survey article [iii] for more details.
Recent work has given us a new line of attack. The cyclotomic quiver Hecke algebras of type A, defined independently (for all oriented quivers) by Khovanov and Lauda and by Rouquier have been shown by Brundan and Kleshchev to be isomorphic to Ariki-Koike algebras, which include the Hecke algebras and the symmetric group algebra as special cases. These algebras are Z-graded and have many other interesting features. An excellent review can be found in the survey article [iv]. This project will focus on using the new techniques available to work on problems which are, or are closely related to, classical problems in modular representation theory.
