Schedule:

We are meeting online from 12:00 pm to 2:00 pm on Mondays.

This is the third seminar on buildings (you can find a link to the previous ones in the references).

Our objective is to understand Tits Indices and the theory of building descent.

Since we have new members that did not participate in the first seminar we are trying to make this more self contained, so we are starting from the theory of Coxeter groups and Spherical buildings, towards the topics we need.

Bibliography:

The main references I use are the following books:

Logbook:

(May 11):

On the positive definite bilinear product: One of the main results of today was the following result:

Proposition:

Let C be a Coxeter Matrix and (W, S) the corresponding Coxeter System, then C is of finite type if and only if W is finite.

In the chat it was remarked that every finite group W has a invariant symmetric bilinear form in the usual way: averaging. If we begin with an inner product then this average is again an inner product that is also invariant. Hence: once you know W finite the standard argument of invariance provides you with the one that makes it of finite type.

On crystallographic types:

Definition:

Let C be a idecomposable matrix. C is crystallographic if it is of finite type and all its entries are integers. In such case we call W = W(C) a Weyl group. Otherwise we say C is of noncrystallographic type.

A better, more geometric, reason for this distinction rather than just talking about rational integer entries comes from the corresponding root system. We explained that the root system is the orbit of the basis elements chosen under the action of W. For finite type this set is finite, and we can create the additive subgroup generated by it inside of V (i.e. the integer linear combinations). It is cristallographic when this turns out to be a lattice of V.

For example, for I2(5) we have it is of noncristallographic type. I show part of the points obtained in this additive subgroup:

(May 4) On the longest word: We discussed how the decomposition as words of the different elements of a Coxeter Groups give paths into the Coxeter Complex.

We discussed two examples that appeared as different Coxeter Groups of rank 3 (i.e. three generators). One of them was spherical and the other was Euclidean, and we made the drawings to see how the order conditions cause the “curving” of the complex.

Based on that I made this drawing that shows the path from the chamber marked 1 to that one marked with the longest element.

The coxeter group I used was $F_4$ whose Coxeter Diagram appears on top. This gives a triangulation of the sphere $S^3$. The longest element has length 24.

Alternative References:

The previous seminars:

Some papers I find interesting or useful as we move along: