09346 - Group Theory

Learning outcomes

At the end of the course the student acquires the fundamental knowledge of Coxeter and reflection groups. The student is able to handle problems in algebra and combinatorial and to compute some combinatorial and algebraic invariants in this mathematical context.  

Course contents

Coxeter groups appears in several domains of mathematics, for example as symmetry groups of regular polytopes, as Weyl groups, in the study of Cluster and Kac-Moody algebras and as reflection groups in euclidean and hyperbolic geometry. 

In this course, we will introduce Coxeter groups emphasizing their combinatorial aspects. 

Program

Reflection and Coxeter groups: reduced expressions, exchange property.

Bruhat order, parabolic subgroups, quotient and interval structure.

Weak order.

Classification of the finite and affine Coxeter groups.

Hecke algebras and Kazhdan-Lusztig polynomials.

Kazhdan-Lusztig cells.

Combinatorial characterizations of some Coxeter groups of finite and affine type. 

Complementary information

The course does not need prerequisites. Elementary notions of group theory and linear algebra will be enough. 

The program of this course is independent but related to that of Representation Theory (96759) and Algebraic Combinatorics (96730).

The courses Algebraic Combinatorics and Group Theory are organized on alternate years. The order in which they are attended is not important. 

Readings/Bibliography

Anders Björner, Francesco Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005.

J. E. Humphreys. Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.

Teaching methods

Every week, 4 hours of frontal lessons. 

Exercise sheets will be weekly assigned. Student's work on the exercises will be fundamental for the understanding of the theory.

Assessment methods

Homewok and usual oral exam.

Office hours

See the website of Riccardo Biagioli