Talk 2: Between primitive and 2-transitive, II: association schemes and coherent configurations

Coherent configurations were introduced by Donald Higman (and the equivalent cellular algebras by Weisfeiler and Leman) as a combinatorial and representation-theoretic tool for the study of permutation groups. However, a special case, association schemes, had arisen much earlier in statistics by Bose and several collaborators, and had been generalised in various ways, notably by Delsarte for applications to coding theory. Every group which is not $2$-transitive acts on a non-trivial coherent configuration. But the same is not true for association schemes. A permutation group is called \emph{AS-free} if it preserves no non-trivial association scheme. Such groups must be primitive, but not too many examples are known. I will discuss the existence or non-existence of AS-free groups in the various O'Nan--Scott classes, and also some generalisations. http://caul.cii.fc.ul.pt/slides/PeterCameron_talk2.pdf/