Alice led the discussion again this week. At the end of the session we realised that this will be the last study group for the year due to various people going away for holidays/conferences/meetings in the lead up to christmas and of course the school’s christmas party. We will resume sometime early in the new year.
We first began by discussing the comments on the post on last week’s study group.
Recall from last week how given a building with apartments isomorphic to the Coxeter complex , we can define the W-distance on chambers. We look again at the chamber system we obtain from the Fano plane.
Let denote differ by a point and denote differ by a line. Let x be the chamber whose line is the orange line and point is the bottom left hand corner of the diagram and y be the chamber whose line is the red line and point is the top of the diagram. Then . There are four chambers which are neighbours of y. These are , which has line the red line and point the point in the middle of the line, which has line the pink line and point the top point, which has line the orange line and point the top point, and which has line the red line and point the bottom right hand corner. Then and . Also and while . Moroever, . Thus we have the property that if is a chamber such that then and if , (where is the length of a reduced expression for g in terms of elements of S), then .
This motivates the following definition of a building: A building of type is a pair with C a set (whose elements are called chambers) and , such that for with the following hold:
- if and only if
- If is a chamber such that then and if , then .
- If then there exists such that and .
This alternative definition of a building yields a chamber system such that the relations are defined by if and only if .
Alice says that this definition is what she uses in her papers on buildings as this is usually easier to work with than the other two definitions.
Given a subset J of S, the residue of type J, denoted , is (using the chamber system definition) the connected component containing c when only taking colours in J. Using the distance W-distance definition, it is the set where .
The usual graph distance between x and y is equal to .
Using this definition we have the Gate property: Given and R a residue, there exists a unique such that for all . Moreover, and . That is there is a shortest path from x to any which goes through c. The element c is called the projection of x onto R.
Let be a thick building with apartment set and let act strongly transitively on , that is acts transitively on . Pick one of these pairs . Let , and , which fixes chamberwise. Then . Moreover, is the reflection of mapping onto its s-neighbour in .
Now given we have . Moreover, if then . Thus makes sense and we can also refer to , and for any .
Now have the following properties:
- and admits a set of generators such that is a Coxeter system.
- for all , that is, the stabiliser of does not stabilise . This follows from the thickness of the building.
- , this is called the Bruhat decomposition of G. Proof: Let . Then there exists an apartment A containing and . By the strong-transitivity, there exists mapping to . Let . Also and so , that is . Hence and so .
- , the stabiliser of the J-residue, is equal to . Such subgroups are called special subgroups.
- if and if .
Now any group G with subgroups B, N satisfying the above 6 properties (where in 5 is defined as the union instead of as the stabiliser of the J-residue), is called a group with a BN-pair. Equivalently we call a Tits system.
In fact, these 6 axioms are not independent and later it was realised that a group with a BN-pair could be defined by the following:
- and admits a set of generators such that for all .
BN-pairs are not unique for a given group G. I suppose an example would be , where the two different interpretations of G as a classical group would give two different buildings for G and so two different BN-pairs. Hopefully Alice can confirm this.
Now starting with a group G with a BN-pair, we can let C be the set of right cosets of B in G. Then we can define if and only if where . (It turns out that the axioms imply that the elements of S are involutions.) This gives a building where one apartment is and the whole set of apartments is .