Math 668: Coxeter groups and root systems

Fall 2024

Course meets: TuTh 1:00-2:20, 3866 East Hall.

Instructor: Sergey Fomin, 4868 East Hall, 764-6297, fomin@umich.edu

Grader: Yuchong Zhang, zongxun@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/668f24.html

Level: introductory graduate.

Student work expected: several problem sets.

Synopsis: Coxeter groups and root systems are simple algebraic/geometric/combinatorial gadgets that arise in multiple mathematical contexts, including representation theory, invariant theory, algebraic geometry, and mathematical physics. This course is a gentle introduction to the subject, emphasizing combinatorial aspects. No graduate-level prerequisites will be assumed.

Tentative list of topics: Classification of finite Coxeter groups. Classification of finite crystallographic root systems. Rings of invariants and coinvariants. Combinatorics of reduced words. Weak and Bruhat orders. Permutohedra and associahedra. Hyperplane arrangements associated with root systems.

Reference texts (none required):

• F. Brenti and A. Björner, Combinatorics of Coxeter groups, Springer-Verlag, 2005.
• J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1994.
• R. Kane, Reflection groups and invariant theory, Springer-Verlag, 2001.
• S. Fomin and N. Reading, Root systems and generalized associahedra, IAS/Park City lecture notes, 2004.