# Cherednik Algebras and Applications Learning Seminar Spring 2021

Last Updated:

### Time and location information

Time: Wednesday at 5:00 - 6:15 pm
Organizer: Sam DeHority
Zoom info: The zoom link is https://columbiauniversity.zoom.us/j/98248875411. Please email me at samdehority@math.columbia.edu to be added to the mailing list and for the zoom password.

### References and topics

The basic plan for the semester is to first cover the fundamentals of the structure of Cherednik algebras focusing on examples and then cover a variety of applications based on the interests of the speaker.

A rough list of topics is as follows:

1. Review of Hecke/ affine Hecke algebras
2. Definition of Cherednik Algebra, PBW theorem, spherical subalgebra
3. Rational and trigonometric degenerations, Dunkl operators, relationship with integrable systems
4. Macdonald polynomials, Macdonald constant term conjecture

For these topics general references are https://arxiv.org/abs/math/0404307 and the associated book, Double affine Hecke algebras by Cherednik, the notes https://arxiv.org/abs/1001.0432, and a recent seminar (elsewhere) with notes https://web.northeastern.edu/iloseu/DAHAEHA.html.

along with many other possibilities and applications depending on interest:

1. Geometric rep theory and affine Springer fibers https://arxiv.org/abs/math/0207127, https://arxiv.org/abs/0705.2691, etc.
2. Relationship with the elliptic hall algebra https://arxiv.org/abs/1202.2756 and see Northeastern/MIT notes above
3. Knot invariants, superpolynomials
4. etc.

### List of talks

Date Talk Info
Wednesday
Feb. 10
Speaker: Álvaro Martínez
Title: Introduction to Hecke algebras and affine Hecke algebras
Abstract: We will motivate the appearance of (affine) Hecke algebras in Lie theory and discuss their basic properties and representations.
Notes: pdf
Wednesday
Feb. 17
Speaker: Henry Liu
Title: DAHAAHAHA!
Abstract: We’ll define a DAHA and its polynomial representation, which will provide some motivation for the D(oubling) and also some structural properties of the algebra.
Notes: pdf
Wednesday
Feb. 24
Speaker: Davis Lazowski
Title: Degenerate DAHA and integrable systems.
Abstract: I will discuss how the rational degeneration of DAHA is related to Olshanetsky-Perelomov Hamiltonians, Bessel functions and Knizhnik-Zamolodchikov equations.
Notes: pdf
Wednesday
Mar. 6
NO TALK due to Spring Break
Wed
Mar. 10
Speaker: Davis Lazowski
Title: Category O for the rational Cherednik algebra
Abstract: We’ll discuss the basic structure of categories O for rational Cherednik algebras, define induction and restriction functors between them, and discuss connections to the KZ equation.
Notes: pdf
Wed
Mar. 17
Speaker: Zoe Himwich
Title: MacDonald Polynomials and the MacDonald Constant Term Conjecture
Abstract: I will introduce MacDonald polynomials. I will discuss properties of these polynomials, and MacDonald’s constant term conjecture.
Thursday
Mar. 25
Speaker: Sam DeHority
Title: DAHA, MacDonald polynomials, and Hilbert schemes
Abstract: We will discuss how various conjectures about MacDonald polynomials are proven using DAHA and the geometry of the Hilbert scheme of points.
Wed
Mar. 31
Speaker: Jin-Cheng Guu
Title: Classical and quantum Schur-Weyl Duality
Abstract: Decomposition of modules is an essential technique in representation theory. For example, given a finite group representation $V \otimes V$, we can always break it down to the symmetric and antisymmetric part. This got generalized to more copies, in the classical Schur-Weyl duality. We will then address its quantum (affine) version. If time permits, we will talk about it in the toroidal setting.
Notes: pdf
Wed
Apr. 7
Speaker: Zoe Himwich
Title: Macdonald Processes
Abstract: I will describe Macdonald processes, sequences of probability measures defined via the Macdonald polynomials. I will also discuss some special cases, such as Schur processes.
Wed
Apr. 14
NO TALK
Wed
Apr. 21
NO TALK
Wed
Apr. 28
Speaker: Sam DeHority
Title: Affine Springer fibers and DAHA
Abstract: Representations of DAHA and its degenerations admit geometric descriptions in terms of cohomology groups of affine Springer fibers.
Notes: pdf