Hmaida, Mufida Mohamed A.
(2016)
*Representation Theory Of Algebras Related To The Bubble Algebra.*
PhD thesis, University of Leeds.

## 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.

## Metadata

Supervisors: | Parker, Alison E. and Martin, Paul P. |
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Keywords: | Bubble algebra, cellular algebra, multi-colour symmetric groupoid and multi-colour partition algebra |

Awarding institution: | University of Leeds |

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.701723 |

Depositing User: | miss M. M. A. Hmaida |

Date Deposited: | 23 Jan 2017 11:47 |

Last Modified: | 11 Mar 2020 10:53 |

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