Watson May, Leyna (2024) Hidden Symmetries, Trees and Operads. PhD thesis, University of Sheffield.
Abstract
We investigate a number of symmetric structures, including spaces of trees, partition complexes, and operads. In particular, these spaces have natural actions of the symmetric group by permuting leaf labels, elements, and labels of inputs to operations respectively. These also share the property that the action of the symmetric group \gls{Sn} may be extended to an action of a larger symmetric group $\Snp$ on arity $n$ objects. In some cases such as the partition complex, this action is not obvious, and is realised through its relationship with the tree space of Robinson and Whitehouse in [RW96]. In the case of operads, the ability to extend the action in this way gives an extra structure to those for which it is compatible with the operad structure. Operads for which this is possible are called cyclic operads, a concept introduced by Getzler and Kapranov in [GK95].
In this thesis we write explicit proofs of equivalences of skeletal and non-skeletal definitions of cyclic operads and cooperads. We explore extended symmetric group actions in the partition poset, on collections of finite sets, and on examples of operads and cooperads with cyclic structure. We give a topological construction using suspensions of tree spaces, that have a variant of a cyclic structure that we introduce. This leads to an anticyclic structure on the desuspension of the Lie operad.
Metadata
Supervisors: | Whitehouse, Sarah |
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Keywords: | Trees, operads, cyclic operads, cooperads, symmetric group, representations |
Awarding institution: | University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Academic unit: | School of Mathematical and Physical Sciences |
Depositing User: | Mrs Leyna Watson May |
Date Deposited: | 14 Jul 2025 16:02 |
Last Modified: | 14 Jul 2025 16:02 |
Open Archives Initiative ID (OAI ID): | oai:etheses.whiterose.ac.uk:37075 |
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