Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.701723
Title: Representation theory of algebras related to the bubble algebra
Author: Hmaida, Mufida Mohamed A.
ISNI:       0000 0004 5993 0958
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2016
Availability of Full Text:
 Access from EThOS: Access from Institution:
Abstract:
In this thesis we study several algebras which are related to the bubble algebra, including the bubble algebra itself. We introduce a new class of multi-parameter algebras, called the multi-colour partition algebra $P_{n,m} ( \breve{\delta} )$, which is a generalization of both the partition algebra and the bubble algebra. We also define the bubble algebra and the multi-colour symmetric groupoid algebra as sub-algebras of the algebra $P_{n,m} ( \breve{\delta} )$. We investigate the representation theory of the multi-colour symmetric groupoid algebra $\F S_{n,m}$. We show that $\F S_{n,m}$ is a cellular algebra and it is isomorphic to the generalized symmetric group algebra $\F \mathbb{Z}_m \wr S_n$ when $m$ is invertible and $\F$ is an algebraically closed field. We then prove that the algebra $P_{n,m} ( \breve{\delta} )$ is also a cellular algebra and define its cell modules. We are therefore able to classify all the irreducible modules of the algebra $P_{n,m} ( \breve{\delta} )$. We also study the semi-simplicity of the algebra $P_{n,m} ( \breve{\delta} )$ and the restriction rules of the cell modules to lower rank $n$ over the complex field. The main objective of this thesis is to solve some open problems in the representation theory of the bubble algebra $T_{n,m} ( \breve{\delta} )$. The algebra $T_{n,m} ( \breve{\delta} )$ is known to be cellular. We use many results on the representation theory of the Temperley-Lieb algebra to compute bases of the radicals of cell modules of the algebra $T_{n,m} ( \breve{\delta} )$ over an arbitrary field. We then restrict our attention to study representations of $T_{n,m} ( \breve{\delta} )$ over the complex field, and we determine the entire Loewy structure of cell modules of the algebra $T_{n,m} ( \breve{\delta} )$. In particular, the main theorem is Theorem 5.41.
Supervisor: Parker, Alison E. ; Martin, Paul P. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.701723  DOI: Not available
Share: