This week's seminar is given by our own Klara Stokes.
Title: Geometries with trialities but no dualities
Abstract: The notion of triality is more elusive than projective duality. It is a phenomenon appearing in a certain 6-dimensional hyperbolic quadric in 7-dimensional projective space, first discovered in the 19th century by Study when he investigated the parametrizations of the rigid motions of 3-dimensional Euclidean space. Nowadays the same triality phenomenon is best known as "an outer automorphism of groups of type D_4, coming from a graph automorphism of the diagram". Actually, this diagram has the symmetric group on 3 elements as automorphism group, so the geometry not only has a triality, but a total of 2 trialities and 3 dualities. In this talk I will show how to get new incidence geometries from embeddings of graphs on surfaces with trialities but no dualities. I will also show how to use Suzuki groups to get an infinite family of flag-transitive incidence geometries with trialities but no dualities. The talk will be adapted to a general audience.
