# Henry Liu

## Modern Techniques in Representation Theory (Spring 2021)

This is an online learning seminar on modern techniques in representation theory, which will include: Soergel bimodules, parabolic category $$\mathcal{O}$$, and KLR algebras.

Talks will roughly be 45 minutes followed by a 5-10 minute break followed by 45 more minutes.

Please email Cailan at ccl at math dot columbia dot edu if you would like to join the seminar, and/or for the password to access the recordings of the talks.

### Rules for the seminar

1. You must have an example/computation in $$\mathfrak{sl}_2$$ or $$\mathfrak{sl}_3$$ in your talk.
2. You cannot give a slides talk unless your talk does not have an example/computation in $$\mathfrak{sl}_2$$ or $$\mathfrak{sl}_3$$.
3. Turn your video on (for the most part) as a courtesy to the speaker.
4. There are no dumb questions.

Please send your title and abstract to Cailan by Wednesday night, and your notes to Henry before your talk starts. If by some miracle you have your notes written and talk prepared by the end of Wednesday, you can include the notes in your email to Cailan.

### Schedule

 Fri Jan 22 Álvaro Martínez Introduction to Soergel Bimodules Soergel bimodules are a combinatorial categorification of the Hecke algebra, and can be used to give an algebraic proof of the Kazhdan-Lusztig conjecture. In this talk we will review preliminary notions such as the Hecke algebra of a Coxeter system, define Soergel bimodules and see some examples of categorification. Links: notes (PDF) and recording Fri Jan 29 Jin-Cheng Guu Pictorial presentation of 2-categories We will define 2-cats $$C$$ and provide some examples. While different presentations of $$C$$ are available, we will use the pictorial presentation as it sheds insight on the structure of $$C$$. Then we will address a special case of 2-cats, the monoidal cats, which are ubiquitous in modern mathematics. A toy but important example is the Temperley-Lieb category. Interestingly, it has even more structures. We will see how those structures mean in the pictorial presentation. Links: notes (PDF) and recording Fri Feb 05 Micah Gay One-Colour Calculus (or: What Diagrammatics Can Do for You, if You’re a Bott-Samelson Bimodule Corresponding to a Simple Reflection) First, we will define Frobenius algebra objects for monoidal categories, and discuss how they arise in the case of Bott-Samelson bimodules. Then we will draw many, many pictures to develop the diagrammatics we discussed last time to describe Bott-Samelson bimodules corresponding to a simple reflection. Links: recording Fri Feb 12 Nikolay Grantcharov Parabolic Category O We introduce parabolic subalgebras and study basic properties of the parabolic analogue of BGG category O. In particular, we give a characterization of objects in parabolic category O in terms of their simple components, and we highlight the main differences with the standard BGG category O by illustrating an example for $$\mathfrak{sl}_3(\mathbb{C})$$. Links: notes (PDF) and recording Fri Feb 19 Álvaro Martínez The Dihedral Cathedral We will develop a diagrammatic presentation for $$\mathbb{BS}\mathrm{Bim}$$ in the case of dihedral groups, this time using two colors. Links: notes (PDF) and recording Fri Feb 26 Cailan Li Singular Soergel Bimodules We introduce Singular Soergel Bimodules and explain the connection with singular Category O. We then go over their diagrammatics in one color and afterwards explain the role they play in the algebraic Satake equivalence, in particular the connection to finite dimensional representations of lie algebras. Links: notes (PDF) and recording Fri Mar 05 Jin-Cheng Guu Koszul duality and Kazhdan-Lusztig conjecture Generalized Koszul dualities show up in many different contexts. Roughly speaking, they interchange simple objects of one category with projective objects of another category. As in the ext-sym example below, it's clear that two categories need not be the same. However, the Kazhdan-Lusztig conjecture (proved) suggests that there should be a Koszul self-duality on category O. It is more subtle however, as such phenomenon appears only when some suitable gradings are remembered (slogan: symmetries that show up only after deformation). In this talk, we will introduce Morita theory in both the classical and the differential-graded context. We will explain that the Koszul duality between the exterior algebra and symmetric algebra is an example of the latter. Time permitting, we will address Koszul duality for category O. Reference: chapters 26 and 27 of Introduction to Soergel Bimodules, 2020 Links: notes (PDF) Fri Mar 12 No seminar Fri Mar 19 Speaker: TBA TBA TBA Notes: TBA Fri Mar 26 Speaker: TBA TBA TBA Notes: TBA