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

If you are interested in joining, please contact me.


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.


The main references I use are the following books:

1 Buildings, Peter Abramenko, Kenneth S. Brown Springer GTM 248
2The Structure of Spherical Buildings, Richard M. Weiss
3The Structure of Affine Buildings, Richard M. Weiss, Annals of Mathematical Studies 168
4Descent in Buildings, Bernhard Mühlherr, Holger P. Petersson, Richard M. Weiss, Annals of Mathematical Studies 190
5Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, Meinolf Geck, Götz Pfeiffer, Oxford Science Publications, New Series 21


DateWhat did we do?NotesBoard
May 4We defined what a Tits Index is to motivate the things we need to study.
We defined spherical Weyl groups and spherical parabolic subgroups and explained briefly how to construct the Coxeter Complex.
[1] Chapter 2 and 3,
[4] Definition 20.1, Chapter 20.
May 11We saw the material on the first 6 pages of the notes.
We introduced Cartan Matrices, standard Cartan matrices, and finite type Cartan matrices as well as the reflection groups associated with them.
We discussed the way to construct Coxeter diagrams and Dynkin diagrams and mentioned, in terms of them, those that correspond to finite type idecomposable ones.
[1] Chapter 1, section 1.1 to 1.3, 1.5.4 to 1.5.6.
[5] Chapter 1, section 1.1 to 1.3
May 18We finished the set of notes of today and those of previous time (what we were missing). We studied the classical types of Coxeter Systems and discussed Graph Automorphisms, Rigid and strongly rigid.
[5] Chapter 1, section 1.4
[Alt 1] For the discussion on rigid, strongly rigid, etc.
May 25We saw the set of notes except for the parts of double cosets. We will return to them later.

We studied algorithmical aspects of Coxeter Systems: computing system of representatives for parabolic subgroups, computing the longest word in spherical buidlings, and discussed the coset graph briefly.

We recalled the concept of M-reduced words and saw how together with folding we can actually make the computations required to do these algorithms.

[1] Chapter 2, Theorem 2.33
[5] Chapter 1, section 1.5
Chapter 2, section 2.1 and 2.2
June 1We discussed conjugacy classes in coxeter classes, in particular the cuspidal ones and how we can understand them via coxeter elements.

[5] Chapter 3, section 3.1.
June 8We began the study of coxeter complexes by reviewing examples and their definition. We mainly discussed how they are constructed and how words correspond to paths through the chambers.

At the end we gave the definition of building whose approach is via simplicial complexes.

[1] Chapter 3, section 3.1, section 3.5. Chapter 4, section 4.1 and 4.2
[4] Chapter 19
June 15We discussed three definitions of buildings and focused mainly in the combinatorial and the geometric one and comparing their axioms and what they mean.

We introduced the notion of residue which will play an important role.

[1] Chapter 4, section 4.8, Chapter 5, section 5.1, 5.2, 5.3.
[4] Chapter 21
June 22We defined residues in a building and stated the gate property. In order to understand its proof we discussed the version of the triangle inequality for the Weyl Distance.

We didn’t finish the proof of the gate property, as we spent almost all class discussing the triangle inequality and what appears in the board is the incomplete proof (just the start…).

[1] Chapter 5, section 5.3. Lemma 5.28 is the triangle inequality and Proposition 5.34 is the Gate property.
[4] Chapter 21, Lemma 21.5 is the triangle inequality and Lemma 21.6 is the Gate property. Thus book doesn’t call them like this.
June 29This class was dedicated to understand in a better way the triangle inequality and the construction of the Weyl distance function of a building from the distance functions of its apartments.

We also discussed how the relationship of the weyl distance and a word decomposition has to do with galleries and homotopies.

In a certain sense, this class’ topics should go before the previous ones but the doubts arised after that class.

[1] Chapter 3, section 3.5 for the construction of the Weyl distance function of a coxeter complex (i.e. an apartment).
Chapter 3, section 3 and chapter 4, section 4.12 to see more about homotopy of buildings.
Chapter 4, section 4.8 for the Weyl distance function in the Building case.
July 6This class got cancelledN/AN/A
July 13We studied residues and projections by stating and proving its main properties. We also defined the notion of the parallel residues.

[1] Chapter 5, section 5.3.2.
[4] Chapter 21, Results from 21.1 to 21.12. In particular, 21.8 and 21.10 are the corresponding properties as astated in this book to what we proved.
July 20We introduced again the notion of Tits Index and some definitions.

We are working toward obtaining the notion of relative Tits index.
In the process we defined: anisotropic, isotropic, split and quasi-split Tits indices.

[4] Chapter 20, Definition 20.1 up to Lemma 20.20.
July 27
August 3
August 10
August 17
August 24
August 31

Comments and Notes:

(May 11):

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


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:


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:

Not a lattice!

(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:

Buildings I
Buildings II(this one is super incomplete, I will be updating it).

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

PaperWhy did I use it?
The isomorphism problem for Coxeter groups, Bernhard MühlherrI was learning about the Twist Conjecture.