On Tuesday, Şükrü gave a talk in our groups and combinatorics seminar on material related to last week’s discussion. The slides of his talk are available on our seminars page and include a definition of a Phan system.
Alice Devillers led the discussion in the study group on Friday to give a buildings perspective on last week’s discussion by Şükrü.
Recall that a chamber system consists of a set of chambers such that two chambers are -adjacent if they agree in all but their elements of type . See also this earlier discussion. We say that a chamber system is simply 2-connected if every closed path can be reduced to the trivial path bya sequence replacing any subpath lying in a rank 2 residue by another path in the same residue, that is, any closed path is 2-homotopy equivalent to the trivial path. Buildings are simply 2-connected chamber systems.
Let be an automorphism group of a chamber system which is transitive on chambers and fix a chamber . Let be the stabiliser in of the set of all chambers which are -adjacent to . This is the stabiliser in of the flag consisting of all the elements of other than the element of type . The group is called a maximal parabolic but it should be noted that it is minimal by inclusion amongst the parabolic subgroups. Given types with we have is the stabiliser of , that is a Borel subgroup . If is connected then where is the set of all types. Moreover, Tits’ Lemma states that if is simply 2-connected then is the 2-amalgam of the maximal parabolics, that is, is generated by the and the only relations in are those which occur in the subgroups for .
For example, in the projective space we can take
Then for we have and
Let be a spherical building, that is, one whose Coxeter system is finite. Since it is finite it has a longest word . We say that two chambers are opposite if , where is the distance defined in the building (see this earlier discussion), that is, if and are as far apart in the building as possible. We can then define the opposite geometry whose chambers are the pairs such that are opposite chambers in . We define adjacency by if and only if . Thus , where is the element of the fundamental system corresponding to type . Then and as we must then have . However, and must be adjacent chambers and so for some . Hence we can define a map on the type set , such that . In particular, . This map gives rise to a symmetry of the Dynkin diagram.
For the Coxeter system of type , induces the reflection of the Dynkin diagram. For and , is the identity. For , if is even then is the identity while if is odd interchanges the two tail nodes of the diagram. Similarly for , if is even is the identity while for odd it interchanges the two nodes of the diagram.
Mühlherr and Abramenko have proved that is simply 2-connected if there are no rank 2 residues isomorphic to the generalised polygons , , or . This result is analogous to the Curtis-Tits Theorem discussed last week so we must have needed the field condition that if there were double bonds in the diagram then the field was not equal to .
We will demonstrate this by looking at the projective space . Let
be an opposite chamber. In general, if and where have dimension , we have that is opposite to if and only if .
Let , which acts on . Then the Borel subgroup is the subgroup of all diagonal matrices, that is, what is usually referred to as a maximal torus . Moreover, the minimal parabolic is equal to the subgroup of all matrices of the form
and so is isomorphic to . Hence it is like one of the copies of from last week. Since is simply 2-connected we have is the 2-amalgam of the minimal parabolics .
The study group will be having a break for a couple of weeks as various key participants will be away.