Oberseminar Algebra und Geometrie

Summer term 2025: Buildings and classical groups and mixed topics

Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.

Organized by Clotilde Gauthier and Alena Ramona Meyer

All talks take place on Fridays at 14:30 in 25.22.03.73.

This term, the oberseminar is divided in two parts. In the first part we will learn about building theory and in the second part we will have mixed topics, mainly with guest speakers. Please get in touch if you have a suggestion on whom to invite.

If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

Infos für Studierende

Im SoSe 2025 wird die erste Hälfte des Seminars aus einer Vortragsreihe über die Theorie der Gebäude bestehen. Die zweite Hälfte wird sich aus Gastvorträgen mit einem losen Bezug zur ersten Hälfte zusammensetzen. Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können, zu promovieren. Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.

Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.

Buildings and classical groups

In order to understand groups, one can construct a building on which the group of interest acts nicely. Indeed one can, for example, retrieve $BN$-pairs of a group or get information about its parabolic subgroups by studying its action on a building. In this seminar, we will first introduce Coxeter groups and chamber complexes in order to then be able to describe building theory in an elementary way (the fact that apartments are Coxeter complexes will be a consequence not an axiom). Then we will explain the link between buildings and $BN$-pairs, and finally we will apply building theory to the general linear group and to isometry groups.

Schedule

11.04.25: Fabian Rodatz: Coxeter groups
This talk is mainly giving definitions and setting the stage for later.
Introduce Coxeter systems, Coxeter groups and Coxeter diagrams. Define reflections and give the key ingredients to establish Tits' linear representation of a Coxeter group (Proposition page 6). Define the set of roots of a Coxeter system and justify that Tits' linear representation is injective. Introduce generalized reflections, give the Strong Exchange Condition and the Deletion Condition, and state equivalent characterization of Coxeter systems. Introduce the Bruhat order and the special subgroups of a Coxeter group, and give their basic properies.
(Source: Chapter 1)
18.04.25: No Seminar
25.04.25: Marcelo Fortmann: Chamber complexes
This talk is also aiming at defining the objects that will appear when working with buildings.
Give the terminology associated to chamber complexes and prove the uniqueness lemma. Define foldings and walls and state some properties. Introduce the Coxeter complex associated to a Coxeter system, state its characterization by foldings and walls, and its consequences.
(Source: Chapter 3)
02.05.25: No Seminar
09.05.25: Doris Grothusmann: Buildings and $BN$-pairs
Give the definitions of apartments and buildings, and explain the canonical retraction to apartments. Prove that apartments are Coxeter complexes. Introduce labels and links and justify that the Coxeter system associated to a building does not depend on the apartment system. Define spherical buildings and give their properties. Define $BN$-pairs and explain how to get a $BN$-pair from a building.
(Source: Chapter 4 and Section 5.1-5.2)
16.05.25: Daniel Echtler: $BN$-pairs
Define parabolic subgroups and generalized $BN$-pairs. State their properties. Detail the spherical case and explain how to get buildings from $BN$-pairs.
(Source: Sections 5.3 to 5.7)
23.05.25: Adrian Baumann: Geometric Algebra
This talk is independent of the previous ones and aims at preparing the next two talks in which we will see applications of building theory.
Define flags, parabolic subgroups, unipotent radical, Levi component and $K$-split torus in the context of $\mathrm{GL}_n$. Summarize the definitions and properties of bilinear and hermitian forms (you can skip pages 108 to 111). State Witt's theorem (Theorem page 111) and explain its consequences on parabolics (Proposition page 115). Give the symplectic groups as an example (Section 8.1).
(Source: Chapter 7 and Section 8.1)
30.05.25: No Seminar
06.06.25: tba: Spherical construction for $\mathrm{GL}_n$
Describe the Coxeter systems of type $A_n$ (First paragraph of Section 2.1). Construct the associated building and describe the action of $\mathrm{GL}_n$ on the building. Describe the spherical $BN$-pairs in $\mathrm{GL}_n$ and prove that, when restricted to $\mathrm{SL}_n$, the action on the building is still strongly transitive. Finally, prove that symmetric groups are Coxeter groups.
(Source: First paragraph of Section 2.1 and Chapter 9)
13.06.25: tba: Spherical construction for isometry groups
Describe the Coxeter systems of type $C_n$ (Second paragraph of Section 2.1). Construct the associated building and describe the action of the group on the building. Describe the spherical $BN$-pairs associated to isometry groups and give an analogue for similitude groups.
If time allows, you can also explain the differences with the oriflamme complexes (Last paragraph of Section 2.1 and Chapter 11).
(Source: Second paragraph of Section 2.1 and Chapter 10)

References:

P. Garrett, Builidings and Classical Groups. Chapman and Hall, London, 1997. link

Mixed Topics

20.06.25: No Seminar
27.06.25: tba
04.07.25: tba
11.07.25: No Seminar
18.07.25: tba