Buildings, Geometries and Algebraic Groups Study Group IV

John led the discussion last week.  I am a bit late in posting as I have been busy organising the WA Junior Olympiad which was on Saturday.

The topic this week is what is called Buekenhout geometries or diagram geometries. We also have a new name.

As we have seen so far, a projective space can be associated with the Dynkin diagram of type A_n and polar spaces can be associated with the Dynkin diagram of type B_n. Tits’ theory of buildings generalises these notions and associates to each simple group of Lie type a geometry known as a building and this building can be encoded in the Dynkin diagram associated with the group. In his seminal 1979 paper, `Diagrams for geometries and groups’ in JCTA, Francis Buekenhout looked to generalise this further and introduced the notion of diagram geometries. These are also referred to as Buekenhout geometries.

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Phan systems study group

(with help from John Bamberg, Alice Devillers, and Sukru Yalcinkaya.)

We have recently started a study group here with the rather ambitious aim of exploring the links between diagram geometries, buildings, Phan systems, groups of Lie type and algebraic groups.  It is very informal, on a Friday afternoon and often involves several people at the blackboard during the hour and plenty of questions. We are also trying to do it via examples. We have decided that it would be a good idea to keep of record here of what we cover. If things go really well one of us will try to blog directly during the study group instead of taking notes. Before we can do so I need to give a brief overview of what we have covered so far.  Future posts should be more fleshed out. We are mainly working from Don Taylor’s book (The Geometry of Classical Groups), Humphreys book (Reflection Groups and Coxeter Groups), Carter’s book (Simple Groups of Lie Type), Abramenko and Brown’s book (Buildings) and Garrett’s book (Buildings and Classical Groups). Comments and corrections are more than welcome.

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