Title:

Representation theory of diagram algebras : subalgebras and generalisations of the partition algebra

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 dtonal partition algebra and the planar dtonal partition algebra. Regarding the dtonal 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 dtonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the dtonal 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 dtonal partition algebra is quasihereditary and generically semisimple. The standard modules of the planar dtonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2tonal partition algebra is closely related to the two coloured FussCatalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour FussCatalan algebra, under certain known restrictions.
