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Coxeter Groups

### Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 12:30–1:45 in MSB 315. The registrar calls this Section 001.

PREREQUISITES: Math 5210 (Graduate Algeba I) or equivalent experience.

MAIN TEXTS: We start out with chapters 1–4 of the text Combinatorics of Coxeter Groups (Springer, 2005) by Anders Björner and Francesco Brenti.

OTHER REFERENCES: Since the theory of Coxeter groups lies at a major intersection of combinatorics, group theory, representation theory, invariant theory, and geometry, there are many other references that provide useful points of view or connections.

• James E. Humphreys: Reflection Groups and Coxeter Groups.
• R. Kane: Reflection groups and invariant theory.

WEBPAGE: The homepage for this course is http://www.math.uconn.edu/~troby/Math5020S16/. Some other web resources are listed below.

SOFTWARE: While doing some computations by hand is generally essential for learning, having software that can do bigger computations or check one's work is very useful. The free open-source computer algebra system Sage has an extensively-developed collection of objects and algorithms implemented for algebraic combinatorics with various documentation including thematic tutorials. More generally, Christian Krattenthaler has kindly compiled a comprehensive list of Combinatorial software and databases.

Specifically for Coxeter groups, the Sage tutorial called Weyl Groups, Coxeter Groups and the Bruhat Order might be a good entry point. A google search on "Sage Coxeter group" also yields some useful pages. For Maple users, John Stembridge's excellent coxeter package is highly recommended.

GRADING: Your grade will be based on several HW assignments, and possibly a presentation given in class.

The text has some good exercises, and I may take some others from various places. You may find some homework problems to be challenging, and collaborating to find solutions is encouraged. While it's OK to get significant help from other resources, in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policy. It also is a poor learning strategy.

ACADEMIC INTEGRITY: Please make sure you are familiar with and abide by The Student Code governing Academic Integrity in Undergraduate Education and Research.

CONTENT: Coxeter groups are abstract groups that can be given a formal presentation in terms of reflections. Finite examples include the symmmetry groups of regular polytopes and the Weyl groups of simple Lie algebras; infinite ones include groups corresponding to regular tesselations of the Euclidean or hyperbolic planes, and the Weyl groups of Kac-Moody algebras. The theory of Coxeter groups is a fundamental and very active area of research, with a beautiful interplay of algebraic, combinatorial, and geometric ideas. The symmetric group is one of the most basic examples; many facts about its structure and representation theory have interesting generalizations to Coxeter groups of other types. Analogues of such basic permutation statistics as the number of inversions or number of descents lead to interesting combinatorial identities. The only prerequisite for this course is a year of graduate abstract algebra (or the equivalent). Notions from combinatorics and representation theory will be developed as needed. After the basic material is covered, the choice of possible topics is quite wide and susceptible to influence by students taking the course.

ACCESSIBILITY & DISABILITY ISSUES: Please contact me and UConn's Center for Students with Disabilities as soon as possible if you have any accessibility issues, have a (documented) disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency.

HOMEWORK ASSIGNMENTS: Homework assignments will be emailed to the class. Some of these problems may be challenging, so don't hesitate to ask for a hint if you're stuck, or submit a partial solution.

• [Due 2/11] HW1
• [Due 3/8] HW2
• [Due 3/30] HW3
• [Due TBD] HW4