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Representations of symmetric groups and related algebras (LYLES_U23SF)

   School of Mathematics

About the Project

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.

Funding Notes

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 View Website
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.


i) G. D. James (1978). The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer.
ii) A. Mathas (1999). Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series 15, American Mathematical Society.
iii) A. Mathas (2004). The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras, Representation theory of algebraic groups and quantum groups, 261-320, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo.
iv) A. Kleshchev (2010) Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. (N.S.) 47 (3), 419-481.

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