|dc.identifier.citation||Speyer, L. 2015. Representation theory of Khovanov–Lauda–Rouquier algebras. Queen Mary University of London.||en_US
|dc.description.abstract||This thesis concerns representation theory of the symmetric groups and related algebras.
In recent years, the study of the “quiver Hecke algebras”, constructed independently
by Khovanov and Lauda and by Rouquier, has become extremely popular. In
this thesis, our motivation for studying these graded algebras largely stems from a
result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras
have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which
generalise the Hecke algebras of type A, and thus the group algebras of the symmetric
groups. This has allowed the study of the graded representation theory of these algebras.
In particular, the Specht modules for the Ariki–Koike algebras can be graded; in
this thesis we investigate graded Specht modules in the KLR setting.
First, we conduct a lengthy investigation of the (graded) homomorphism spaces
between Specht modules. We generalise the rowand column removal results of Lyle and
Mathas, producing graded analogues which apply to KLR algebras of arbitrary level.
These results are obtained by studying a class of homomorphisms we call dominated.
Our study provides us with a new result regarding the indecomposability of Specht
modules for the Ariki–Koike algebras.
Next, we use homomorphisms to produce some decomposability results pertaining
to the Hecke algebra of type A in quantum characteristic two.
In the remainder of the thesis, we use homogeneous homomorphisms to study some
graded decomposition numbers for the Hecke algebra of type A. We investigate graded
decomposition numbers for Specht modules corresponding to two-part partitions. Our
investigation also leads to the discovery of some exact sequences of homomorphisms
between Specht modules.||en_US
|dc.description.sponsorship||Queen Mary University of London; Queen Mary’s Eileen Colyer Prize
and the Australian Research Council grant DP110100050 “Graded representations of
Hecke algebras”, both of which supported my visit to Sydney.||en_US
|dc.publisher||Queen Mary University of London||en_US
|dc.subject||Quiver Hecke algebras||en_US
|dc.title||Representation theory of Khovanov–Lauda–Rouquier algebras.||en_US
|dc.rights.holder||The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author||