Abstract:
Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Marina Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory.
The association scheme of the symmetric group is well known. We will introduce it, and then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these algebraically in the appropriate association scheme . We will give existence results showing that there exist transitive sets of generalised permutations that are small compared to the size of the group. Many of these results extend results previously known for the symmetric group.
No particular knowledge of association schemes will be required to appreciate this talk.
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