Topics in Algebraic Combinatorics
Spring term 2014
General
Information
- 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.