The representation theory of the symmetric group algebra is a classical area of study.  In characteristic 0, the irreducible modules are well-understood, although there remain open problems that are easy to state.  The decomposition of the tensor product of two irreducible modules has only been determined very recently, and it is still not known if Foulkes's Conjecture holds.  In characteristic p, much less is known; in fact, the dimensions of the irreducible modules are still unknown in general.   A framework for working in the area came from the 1976 book [1].  The methods developed were largely combinatorial, involving partitions, tableaux and indeed the abacus. 

In fact, the symmetric group algebra is just one member of a family of algebras known as Ariki-Koike algebras, or cyclotomic Hecke algebras of type G(r,1,n).  The representation theory of these algebras can also be studied using combinatorics, where the Specht modules, an important class of modules, are now indexed by multipartitions.  A survey of these algebras can be found in [2].  However, in 2009, Brundan and Kleshchev showed that the Ariki-Koike algebras are in fact isomorphic to certain cyclotomic quiver Hecke algebras introduced independently by Khovanov and Lauda and by Rouquier [4].  These quiver Hecke algebras are graded algebras and the grading appears to carry important information. 

This project will work with the cyclotomic quiver Hecke algebras of type A [3].  It is convenient to represent elements of the algebra as diagrams, using the diagram calculus introduced by Khovanov and Lauda, and again, combinatorial methods are often used.  This is a new and exciting area of research; and yet it contains as a special case the classical example of the symmetric group algebra.

This project is also open to applicants (home, EU or Overseas) who have their own funding.  Self-funding applicants may also apply to study part-time.

G.D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., vol. 682, Springer-Verlag,  New York, 1978.

A. Mathas, The representation theory of the Ariki–Koike and cyclotomic q-Schur algebras, in: Representation  Theory of Algebraic Groups and Quantum Groups, in: Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo,  2004, pp. 261–320.

A. Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bull American Math. Soc. 47 (2010) 419-481. 

J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009) 451 - 484.