On the combinatorics of set families

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