A permutation group is primitive if it preserves no non-trivial partitions. If a group is imprimitive, then there must exist a partition which is invariant under the action of the group. Such a partition may be thought of as a "witness" to imprimitivity. We can further categorise primitive groups according to the synchronisation hierarchy by considering different witnesses consisting of partitions, sets, and multisets. The ultimate goal, initiated by Araújo, Cameron, and Steinberg, is to classify groups in the synchronisation hierarchy.
A transitive group naturally produces an association scheme, called the Schurian scheme, with relations formed by the orbitals. In this talk I will show that witnesses in the synchronisation hierarchy can be studied as Delsarte designs of the Schurian scheme. Using this framework, we find the first examples of synchronising groups of diagonal type, (almost) classify the separating rank $3$ groups, and pioneer an effective technique for determining if a group is spreading. Much of this work is joint with John Bamberg, Michael Giudici and Gordon Royle.

開催方法: 対面と Zoom によるハイブリッド
開催場所: 6階608演習室
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