Decomposition matrices for symmetric groups and related algebras (LYLES_U18SF)
This PhD project considers modular representations of certain algebras which generalise the symmetric group algebra. 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. However, representations of the symmetric group over fields of positive characteristic are not well-understood: even though it is possible to construct the irreducible modules 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 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 PhD 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. The most important such problem is to understand the decomposition matrices, that is, to find the composition factors of the Specht modules. There exist certain families of (multi)-partitions where this has been shown to be possible; and we will investigate other such families.
This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at http://www.uea.ac.uk/study/postgraduate/research-degrees/fees-and-funding.
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. The amount charged annually will vary considerably depending on the nature of the project and applicants should contact the primary supervisor for further information about the fee associated with the project.
This job comes from a partnership with Science Magazine and