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Title: On the combinatorics of set families
Author: Langworthy, Andrew
ISNI:       0000 0004 7427 0310
Awarding Body: University of East Anglia
Current Institution: University of East Anglia
Date of Award: 2018
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This thesis concerns the combinatorics and algebra of set systems. Let V be a set of size n. We define a vector space Mn with basis the power set of V. This space decomposes into a direct sum of eigenspaces under certain incidence maps. Any collection of k-sets S embeds naturally into this space, and so decomposes as a sum of eigenvectors. The main objects of study are the lengths of these eigenvectors, which we call the shape of S. We prove that the shape of S is a linear transformation of the inner distribution, and show that t-designs have a specific shape. We give some classifications of the shape of collections of k-sets for small k. Given a permutation group G, we define the subspace MG of Mn of all vectors fixed by G. We show that this space is spanned by the G-orbits of the power set of V and as a consequence of this, prove the Livingstone-Wagner Theorem. We then give some results about groups that have the same number of orbits on 2-sets and 3-sets.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available