# 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:

- Review of Hecke/ affine Hecke algebras
- Definition of Cherednik Algebra, PBW theorem, spherical subalgebra
- Rational and trigonometric degenerations, Dunkl operators, relationship with integrable systems
- 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:

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