Summer School on Algebraic and Enumerative Combinatorics

Lectures by Francesco Brenti

S. Miguel de Seide, Portugal

Topic: Combinatorics of Coxeter Groups

• Coxeter systems.
• Length function.
• The geometric representation.
• Root systems.
• Parabolic subgroups and quotients.
• Geometric interpretation of the length function.
• Roots and reflections.
• The exchange condition.
• Bruhat order.
• Reduced expressions and subexpressions.
• Irreducible systems.
• The classification theorem.
• Hecke algebras.
• R-polynomials.
• Kazhdan-Lusztig polynomials.

Suggested reading

• A. Bjorner, F. Brenti, "Combinatorics of Coxeter Groups",
  Graduate Texts in Mathematics, 231, Springer-Verlag, New York, 2005. (Chapter 2)
• J. E. Humphreys, "Reflection Groups and Coxeter Groups",
  Cambridge Studies in Advanced Mathematics, 29, Cambridge Univ. Press, Cambridge, 1990. (Chapter 5)