| dc.description.abstract | The Khovanov{Lauda{Rouquier algebras Rn are a relatively new family of Z-graded
algebras. Their cyclotomic quotients R
n are intimately connected to a smaller family
of algebras, the cyclotomic Hecke algebras H
n of type A, via Brundan and Kleshchev's
Graded Isomorphism Theorem. The study of representation theory of H
n is well developed,
partly inspired by the remaining open questions about the modular representations
of the symmetric group Sn.
There is a profound interplay between the representations for Sn and combinatorics,
whereby each irreducible representation in characteristic zero can be realised as a Specht
module whose basis is constructed from combinatorial objects. For R
n , we can similarly
construct their representations as analogous Specht modules in a combinatorial fashion.
Many results can be lifted through the Graded Isomorphism Theorem from the symmetric
group algebras, and more so from H
n , to the cyclotomic Khovanov{Lauda{Rouquier
algebras, providing a foundation for the representation theory of R
n .
Following the introduction of R
n , Brundan, Kleshchev and Wang discovered that
Specht modules over R
n have Z-graded bases, giving rise to the study of graded Specht
modules. In this thesis we solely study graded Specht modules and their irreducible
quotients for R
n . One of the main problems in graded representation theory of R
n , the
Graded Decomposition Number Problem, is to determine the graded multiplicities of
graded irreducible R
n -modules arising as graded composition factors of graded Specht
modules.
We rst consider R
n in level one, which is isomorphic to the Iwahori{Hecke algebra of
type A, and research graded Specht modules labelled by hook partitions in this context.
In quantum characteristic two, we extend to R
n a result of Murphy for the symmetric
groups, determining graded ltrations of Specht modules labelled by hook partitions,
whose factors appear as Specht modules labelled by two-part partitions. In quantum
characteristic at least three, we determine an analogous R
n -version of Peel's Theorem
for the symmetric groups, providing an alternative approach to Chuang, Miyachi and
Tan.
We then study graded Specht modules labelled by hook bipartitions for R
n in level
two, which is isomorphic to the Iwahori{Hecke algebra of type B. In quantum characterisitic
at least three, we completely determine the composition factors of Specht modules
labelled by hook bipartitions for R
n , together with their graded analogues. | en_US |
