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Representation Theory of Diagram Algebras: Subalgebras and Generalisations of the Partition Algebra

Ahmed, Chwas Abas (2016) Representation Theory of Diagram Algebras: Subalgebras and Generalisations of the Partition Algebra. PhD thesis, University of Leeds.

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Abstract

This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.

Item Type: Thesis (PhD)
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.701726
Depositing User: Mr C. A. Ahmed
Date Deposited: 24 Jan 2017 10:33
Last Modified: 25 Jul 2018 09:53
URI: http://etheses.whiterose.ac.uk/id/eprint/15997

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