Math 765: Buildings and Classical Groups

Announcements:

• 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.