Reading Group — Affine Weyl Groups 2024

This seminar will provide an elementary and combinatorial introduction to affine Weyl groups. These groups are a family of infinite groups that exhibit similar properties to finite Weyl groups. The infinite nature of affine Weyl groups adds a layer of complexity and interest to the theory.

Affine Weyl groups are connected to various research topics in combinatorics, algebraic geometry, and representation theory. Some examples include:

• Representations of affine Lie algebras
• Quantum cohomology of flag varieties
• Macdonald polynomials

The goal of this seminar is to make papers in these areas more accessible to participants. Attendees should have a basic understanding of finite Weyl groups. We recommend the following references to help you prepare:

• Humphreys. Reflection Groups and Coxeter Groups.
• Bjorner, Brenti. Combinatorics of Coxeter Groups.

We will begin with a review of the finite theory in the first lecture. During the seminar, we will refer to the following notes and texts:

• Humphreys. Reflection Groups and Coxeter Groups.
• Macdonald. Affine Hecke Algebras and Orthogonal Polynomials.