Topics in Algebraic Combinatorics

• Instructor: Anders Björner, bjorner(at)math.kth.se
• Teaching Assistant: Afshin Goodarzi, af.goodarzi at gmail.com
• Level: Graduate
• Prerequsites: Basic courses in algebra and combinatorics. Mathematical maturity.
• Place: Seminar Room 3721, Mathematics Dept, KTH, Lindstedtsvägen 25, 7th floor
• Time: Tuesdays 15:15 - 17:00 and Thursdays 10:15 - 12:00.
• Start: Thursday February 6 at 10:15
  
• Examination: In addition to regular participation at the lectures there will be two take-home exams during the course.
  

Course content

Coxeter groups are at the intersection of many branches of mathematics. Loosely speaking, they are the discrete groups generated by reflections. This is a cause for their geometric beauty but does not give even a hint of the central importance in mathematics of this class of groups; in Lie theory (as Weyl groups of root systems), in the geometry of matrix groups (via the theory of BN-pairs), in algebraic geometry (as indexing sets for cell decompositions of grassmannians and flag varieties), in combinatorics (providing a general framework for permutation structures and important posets), and even in analysis (Arnold's classification of singularities).

In the course we will spend about half the time on the general theory of Coxeter groups, touching on topics of combinatorial interest, such as Bruhat partial order and the finite automata for solving word problems. The rest of the time we will visit some related topics, such as groups with BN-pairs and their corresponding geometries -- the Tits buildings, and finally give a glimpse of Kazhdan-Lusztig theory.

References

The following books are recommended for side reading.


• P. Abramenko and K.S. Brown, Buildings, Springer, 2011.
  
• A. Björner and F. Brenti, Combinatorics of Coxeter groups, Springer, 2005.
• J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, 1990.