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

and 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.

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