Coxeter groups arise in nature, whether as symmetry groups of regular polytopes, tesselations of the plane, juggling patterns, or more generally, as reflections. We will look at Coxeter groups from a combinatorial, geometric and algebraic approach.

This course serves both as an introduction to Coxeter groups and a glimpse into the current research trends, including combinatorial descriptions of finite and affine Weyl groups, the complete cd-index and Kazhdan-Lusztig polynomials.

We will be following Björner and Brenti's recent book on Coxeter groups. We will also be reading some recent papers in the field.

TEXTBOOK

COURSE OUTLINE

• Introduction.
• Bruhat order.
• Weak order and reduced words.
• Root systems, games and automata.
• Combinatorial descriptions.
• Enumeration.
• Kazhdan-Lusztig and R-polynomials.
• Other material, as time permits.