Math 665: Combinatorics of Root Systems
Fall 2005
Course meets: TuTh 1:10-2:30, 866 East Hall.
Instructor:
Sergey Fomin, 2858 East Hall, 764-6297,
fomin@umich.edu
Course homepage: http://www.math.lsa.umich.edu/~fomin/665f05.html
Level: introductory graduate.
Student work expected: several problem sets.
Synopsis (tentative):
I. Fundamentals
- Coxeter groups
- Root systems and reflection groups
- Cartan-Killing classification
- Polynomial invariants
- Coinvariant algebra. Schubert polynomials
II. Combinatorial constructions associated with root systems
- Posets: root poset, Bruhat order, weak order, noncrossing
partitions
- Hyperplane arrangements: Coxeter, Catalan, Linial, Shi, etc.
- Polytopes: permutohedron, associahedron, etc.
- Cluster combinatorics
Texts.
The course will not follow a particular text.
Principal sources for Part I:
- F.Brenti and A.Björner, Combinatorics of Coxeter
groups,
Springer-Verlag, 2005.
- N.Bourbaki, Lie groups and Lie algebras. Chapters 4-6,
Springer-Verlag, 2002.
- H.Hiller, Geometry of Coxeter groups , Pitman Publishing, 1982.
- J.E.Humphreys, Reflection groups and Coxeter groups,
Cambridge University Press, 1994.
- R.Kane, Reflection groups and invariant theory,
Springer-Verlag, 2001.
Some of the sources for Part II (see also the papers
cited or linked to therein):
- S.Fomin and N.Reading,
Root systems and generalized associahedra, IAS/Park City lecture notes,
2004.
- R.P.Stanley,
Hyperplane arrangements, IAS/Park City lecture notes,
2004.
- A.Postnikov,
Permutohedra, associahedra, and beyond.
-
Resources on associahedra and noncrossing partitions,
compiled by J.McCammond.
-
Bibliography on cluster algebras,
compiled by F.Chapoton.