Flag Varieties in Representation Theory

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Christian Steinert

Name

Dr. Christian Steinert

Postdoc

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In the summer term, the lecture Flag Varieties in Representation Theory will be given by Christian Steinert for 9 credit points (6h per week). The lecture is designed for Master students and advanced Bachelor students of mathematics and is therefore given in English. The content of the lectures Computeralgebra or Algebra is required.

A flag variety is a set of increasing chains (flags) of subspaces of a given vector space. These varieties are some of the most studied and best understood algebraic varieties. Their geometry admits numerous applications, for example in representation theory.

After a brief introduction into the representation theory of the symmetric group and the general linear group (Schur–Weyl duality), we will encounter different concepts connected to this topic, like the famous theorem by Borel–Weil–Bott, Grassmannians, Schubert varieties, root combinatorics, combinatorics of Young tableaux, the fundamental theorems of classical invariant theory, cohomology and singularities. If time permits, we will shed some light onto current areas of research, such as standard monomial theory, resolution of singularities, Newton–Okounkov theory and toric degenerations.

Previous knowledge about varieties or representations is not required.

Each week covers a section of the script and contains one exercise and two lecture meetings. The respective contents will be acquired during the first lecture and in self-studies using the script, deepened in small groups during the exercise meeting and finally reinforced through questions and discussions in the second lecture meeting.

Lecture meetings will take place Tuesdays and Wednesdays from 16:30 to 18:00 (location: see RWTHonline). Exercise meetings are scheduled for Wednesdays 18:30 to 20:00 but can be moved according to demand.

Registration will be done via RWTHonline at the beginning of the semester. Upon registration one automatically enters the learning space in RWTHmoodle, where exercises will be organized and further information can be found.