- The first lecture will be on 21 September 2006 Thursday between 1.40 and 3.40 in M-215.
- The first meeting is on 19 September 2006 Tuesday at 2 pm in M-214 or 215.
- The course is open now as Math 765.

**What?** Linear reflection groups; affine reflection groups.
Coxeter groups; Coxeter complexes. Spherical buildings; groups with a
BN-pair. Classical groups.

**When?** Monday 12.40-13.30 (M-214)

Wednesday 10.40-12.30 (M-231)

**Main Books.**

- P. Garrett, Buildings and Classical Groups, Chapman Hall, 1997.
- K. Brown, Buildings, Springer, 1989.
- J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.

**Other Books.**

- M. Ronan, Lectures on Buildings, Academic Press, 1989.
- R.M. Weiss, The Structure of Spherical Buildings, Princeton, 2004.
- R. Carter, Simple Groups of Lie Type, John Wiley & Sons, 1972.
- A.V. Borovik and A. Borovik, Mirrors and Reflections, Springer, 2006.
- D. Taylor, The Geometry of Classical Groups, Heldermann, 1992.
- N. Bourbaki, Lie Groups and Lie Algebras, Addison Wesley, 1975.
- J. Tits, Buildings of Spherical Type and BN-pairs, Lecture Notes in Mathematics 386, Springer, 1974.

**In the Class.**

**Week 1:** (2 hours) Introduction to finite
reflection groups. Root systems,
simple systems, simple reflections.

**Week 2:** The deletion condition. An equivalent condition for being
a Coxeter group. Coxeter graphs. Classification of finite reflection
groups.

**Week 3:** A quick review of affine reflection groups.
The geometrization of Coxeter groups.

**Week 4:** Generalized reflections. Exchange and Deletion
Conditions. Bruhat ordering in Coxeter groups.

**Week 5:** Parabolic subgroups in Coxeter groups. Chamber complexes
and foldings.

**Week 6:** No lectures, because of the holiday.

**Week 7:** Coxeter complexes.

**Week 8:** No lectures, because of a conference.

**Week 9:** Characterization of Coxeter complexes by walls.
(Presentation by Erkan M.
T�rkan)

**Week 10:** Characterization of Coxeter complexes by walls
(continued).
(Presentation by Yal��n Karata�)

**Week 11:** (2 hours) Buildings (definitions, examples,
preliminaries).

**Week 12:** Labels, apartment systems, spherical buildings.

**Week 13:** A group of label-preserving simplicial isomorphisms
acting stongly transistively on a (thick) building is a BN-pair.

**Week 14:** Parabolic subgroups, generalized BN-pairs.
(Presentation by G�khan Benli)

**Week 15:** Parabolics and Levi components in the spherical case.
(Presentation by G�khan Benli)

**Week 15+1:** Buildings from BN-pairs (Presentation by Hakan
G�nt�rk�n)

*This line last updated on 13 January 2007.*