Homomorphisms between Specht modules of KLR algebras

Witty, George (2020) Homomorphisms between Specht modules of KLR algebras. Doctoral thesis, University of East Anglia.

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This thesis is concerned with the representation theory of the symmetric groups and related algebras, in particular the combinatorics underlying the representations of the Khovanov-Lauda-Rouquier (KLR) algebras. These algebras are of particular interest since they possess cyclotomic quotients which were shown by Brundan and Kleshchev to be isomorphic to the Ariki-Koike algebras. The Ariki-Koike algebras generalise Iwahori-Hecke algebras of the symmetric group, and so in turn generalise the symmetric groups themselves. Via this isomorphism, we are able to utilise the grading of the KLR algebras in the setting of the Ariki-Koike algebras, and thus study graded Specht modules.

Specht modules of the KLR algebras admit a definition which lends them well to diagrammatic combinatorics. We shall first develop an arsenal of combinatorial lemmas related to the manipulation of braid diagrams. Then, we will use these to demonstrate the existence of explicit homomorphisms between Specht modules of certain KLR algebras, related to moving particular shapes between the multipar-titions associated to these Specht modules. We shall begin by considering moving single nodes between bipartitions, but eventually consider moving multiple large connected shapes of nodes between components of multipartitions in higher levels.

We will then use the obtained homomorphisms to investigate the homomor-phism spaces between Specht modules that lie in core blocks of level 2 KLR algebras whose base tuples consists entirely of zeroes. We will completely describe the dominated homomorphism spaces between Specht modules in these blocks. In particular, when the quantum characteristic is not 2 and the multicharge entries are distinct, we will completely describe all homomorphism spaces between Specht modules in these blocks. We will also give a conjecture about replacing the base tuple with any arbitrary base tuple.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Chris White
Date Deposited: 14 Apr 2021 14:26
Last Modified: 14 Apr 2021 14:26
URI: https://ueaeprints.uea.ac.uk/id/eprint/79754

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