This is a continuation of last semester's learning/research seminar on representation theory, often with an eye towards categorification. Talks will likely be disconnected talks from across representation theory and categorifcation, reflective of what the speaker is currently interested in working on/exploring; in particular this seminar is not cumulative and you are welcome to attend whatever talk you find interesting. There may be talks relating to affine Hecke algebras, tensor categories, Brauer categories, Soergel bimodules, etc..

Talks will roughly be 45 minutes followed by a 5 minute break, followed by 45 more minutes. Please email Cailan at ccl2166@columbia.edu if you'd like to give a talk.

- Organizers: Cailan Li, Alvaro Martinez, and Fan Zhou
- When: Wednesday 4:40pm -- 6:10pm (or possibly longer)
- Where: Math Room 528

- 01/24
- Speaker: Cailan Li
- Title:
**Kazhdan-Lusztig Theory and Highest-Weight Categories**

Abstract: We define abstract Kazhdan-Lusztig theories for highest weight categories following [CPS]. We then give two applications, the first allows us to calculate dimensions of Ext groups between any two simple modules in our category (in particular for Category O). The second gives us a criteria for complete reducibility via "parity" considerations and was one of the motivations for the notion of "parity sheaves." (cf. this and this and this) - 01/31
- Speaker: Fan Zhou
- Title:
**Representation Theory of Quivers**

Abstract: We discuss the representation theory of quivers (with relations), pointing out the constructions of the simples and the projectives/injectives. Every basic finite-dimensional algebra over an algebraically closed field is isomorphic to the path algebra of a quiver, and we say some words on how to find this quiver. - 02/07
- Speaker: Alvaro Martinez
- Title:
**SL_2-Plethysms, Old and New**

Abstract: Much of the history of representation theory and algebraic combinatorics has been concerned with the study of plethysms, which are compositions of Schur functors. Finding formulas for these has been one of the major open problems in mathematics. When evaluated on a representation of $SL_2$, the situation becomes more tractable, yet it is still very rich, especially over arbitrary fields. We will give an overview of the classical results, which go back to the mid-19th century, up to some closely related recent developments. - 02/14
- Speaker: Cailan Li
- Title:
**Clifford Theory for Crossed Product Algebras**

Abstract: Given an action of a group $G$ on an algebra $R$, one can form the crossed product algebra $R\rtimes G$. In preparation for future talks on representations of affine Hecke algebras, we will go over Clifford Theory for Crossed Products algebras, which allows us to construct the simples for $R\rtimes G$ or $R^G$ from the simples of $R$ and $G$. Time permitting, we might talk a bit on classical Clifford Theory for finite groups. - 02/21
- Speaker: Pavel Turek
- Title:
**Schur functors and plethysms in the stable module category of SL_2(F_p) in characteristic p**

Abstract: Schur functors are endofunctors in categories of modules of an algebra. They can be used to construct all polynomial representations of GL(V) in characteristic 0 and to recover Schur polynomials by considering formal characters. A still unanswered question asks to describe compositions of Schur functors, the so-called modular plethysms. We consider this question for the natural two-dimensional module of SL_2(F_p) in characteristic p when the modular plethysms behave particularly nicely and classify ‘small’ modular plethysms which are projective, respectively, projective after forgetting one indecomposable summand. Using endotrivial modules we then find the stable representation ring of SL_2(F_p) and describe how Schur functors act on indecomposable modules of SL_2(F_p). - 02/28
- Speaker: Chris Bowman
- Title:
**Combinatorics and Stabilities of Plethysms**

Abstract: We introduce plethysm as a product on symmetric functions and on representations of symmetric groups. We wish to understand these products combinatorially, and we highlight a few recent results in this direction — in particular, we discuss how certain stabilities allow us to compare and understand these products under certain limits. If time allows, we might discuss how these products appear in geometric complexity theory. - 03/06
- Speaker: Fan Zhou
- Title:
**Koszul Duality and Reconstruction**

Abstract: We briefly explain some aspects of Koszul theory, define the Koszul duality functor (as constructed in Beilinson-Ginzburg-Soergel), and explain its relation to ``reconstruction theory''. This machinery produces concrete results, such as BGG resolutions and standard/Verma filtrations of projectives. In some sense, computing this Koszul duality can be as hard as the Kazhdan-Lusztig problem, and computing BGG resolutions *is* computing this Koszul duality. We will focus on (the central block of) category O as a working example. Time permitting, we may discuss the categorification of Chebyshev/Hermite polynomials as another example. - 03/13
- Speaker: SPRING BREAK
- Title:

Abstract: - 03/20
- Speaker: Jon Brundan (1pm EST)
- Title:
**Odd Grassmannian bimodules**

Abstract: The 2-category of Grassmannian bimodules consists of some particular singular Soergel bimodules which can be used to categorify finite dimensional irreducible representations of sl_2. There is also a remarkable complex of Grassmannian bimodules, the Chuang-Rouquier complex, was introduced originally in order to construct derived equivalences between blocks of symmetric groups. In this talk, I will talk about the odd analog of some these ideas, based on my joint work with Kleshchev. - 03/27
- Speaker: Ulrich Thiel
- Title:
**The center of the asymptotic Hecke category and unipotent character sheaves**

Abstract: Lusztig's asymptotic algebra is a truncation of the usual Hecke algebra for a Coxeter group in a two-sided Kazhdan--Lusztig cell. Lusztig noted in 2015 that this algebra admits a categorification, given by a truncation of the Hecke category (category of Soergel bimodules). We call this the asymptotic category. For a Weyl group, Lusztig showed that the Drinfeld center of the asymptotic category is equivalent to the category of unipotent character sheaves (supported on the fixed cell) on a corresponding reductive group. Consequently, the S-matrix of the center equals the Fourier transform matrix. Subsequently, Lusztig conjectured that for a non-crystallographic finite Coxeter group the S-matrix should match the "exotic" Fourier transform constructed in the 1990s. We proved this conjecture for the dihedral groups and some (we cannot resolve all) cases of H3 and H4. This is joint work with Liam Rogel. - 04/03
- Speaker: Che Shen
- Title:
**Introduction to affine Hecke algebra**

Abstract: I will explain the equivalence between two different presentations of affine Hecke algebra, one due to Iwahori-Matsumoto and another due to Bernstein, and deduce some basic properties from them. Time permitting, I will define the graded Hecke algebra whose representation theory is closely related to the affine Hecke algebra via Lusztig’s reduction theorem. - 04/10
- Speaker: Linliang Song
- Title:
**Categorification related to cyclotomic (oriented) Brauer categories**

Abstract: Ariki’s categorification theorem connects the various fundamental problems in representation theory of cyclotomic Hecke algebras to important invariants of the integrable highest weight module of (affine) type A. This talk establishes an analog of Ariki’s categorification theorem related to cyclotomic oriented Brauer categories and Kauffman categories. These categories are categorical versions of (cyclotomic walled ) Brauer algebras, cyclotomic BMW algebras which appear naturally in Lie theory via Schur-Weyl duality. In particular, the categorification theorem for cyclotomic oriented Brauer category was conjectured by Brundan-Comes-Nash-Reynolds in 2013. The proof relies on the notion of weakly triangular decomposition, which provides a sufficient condition to show that the corresponding categories are upper finite fully stratified category in the sense of Brundan-Stroppel. This talk is based on joint works with H.Rui, and joint works with M.Gao and H.Rui. - 04/17
- Speaker: Cailan Li
- Title:
**Affine Hecke Algebras: Representations**

Abstract: We start by proving Bernstein's theorem on the center of the Affine Hecke Algebra. Then we classify all irreducible representations for the affine hecke algebra in rank 1. Not sure what we will do next as I haven't gotten that far yet. Come at your own risk. - 04/24
- Speaker: Che Shen
- Title:
**Hecke algebra and equivariant K theory of Steinberg variety**

Abstract: I will first explain the isomorphism between affine Hecke algebra and the convolution algebra of Steinberg variety. I will then talk about how to classify all simple modules of affine Hecke algebra using this isomorphism together with a sheaf-theoretic argument. - 05/15
- Speaker: Roy Forestano
- Title:
**Machine Learning Symmetries**

Abstract: We design a machine learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the generators of symmetry transformations. We construct loss functions which ensure that the applied transformations are symmetries and that the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by a symmetry. We learn the (sub)group structure of the rotation groups SO(n), Lorentz group SO(1,3), and unitary groups U(n) and SU(n). Using accelerated algorithms, we derive the exceptional groups G2, F4, and E6. Other non-linear examples include squeeze mapping, piece-wise discontinuous, and MNIST latent space classifier symmetries which demonstrates that our method is completely general with many possible applications.