## Defining an affine partition algebra

Creedon, S. (2022).
*Defining an affine partition algebra*.
(Unpublished Doctoral thesis, City, University of London)

## Abstract

This thesis is focused on constructing an aﬃne version of the partition algebra and investigating expected properties that such an algebra should admit. In fact we provide two deﬁnitions of such an algebra in this thesis. The ﬁrst deﬁnition is given by generators and relations, and comes about by aﬃnizing the partition algebra in a similar manner employed by others on related diagrams algebras. The second deﬁnition is obtained by generalising the orbit basis of the partition algebra in a particular manner. Both such algebras give rise to actions on a tensor space which extends the action in Schur-Weyl duality between the partition algebra and group algebra of the symmetric group. We establish a strong connection to one of these aﬃne partition algebras with the Heisenberg category. Namely we prove that a certain endomorphism algebra of a given object in the Heisenberg category is a quotient of the aﬃne partition algebra.

Pursuing the construction of such algebras has also lead to new results regarding both the partition algebra and symmetric group. For the partition algebra we obtain a complete description of the center in the semisimple case, and give an alternative description of the blocks in the non-semisimple case. For the symmetric group, we generalise certain results regarding the centers of the group algebras of the symmetric groups to certain centraliser algebras. From such we are able to provide a centraliser construction of the degenerate aﬃne Hecke algebra, and show that a certain limit of centralizer algebras appears as an endomorphism algebra in the Heisenberg category.

Publication Type: | Thesis (Doctoral) |
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Subjects: | Q Science > QA Mathematics |

Departments: | School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses |

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