Representation Theory and Categorification Seminar (Fall 2024)

This is a continuation of last year'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, KLR(W?) algebras, quantum affine algebras, Soergel bimodules, etc..

Talks will roughly be 1.25-1.5 hours, e.g. 45 minutes followed by a 5 minute break, followed by 45 more minutes. Please email Fan or Alvaro at fz2326@columbia.edu or alm2297@columbia.edu if you'd like to give a talk!

Schedule

09/05
Speaker: Everyone
Title: Logistics/Information
Abstract: Just for headcount and gauging where participants' interests are.
09/12
Speaker: Igor Chaban
Title: Introduction to (type A) quantum groups
Abstract: I will give the Drinfeld-Jimbo definition of a quantum group, corresponding to a simple Lie algebra. It will be shown that it has a Hopf algebra structure providing a 1-parameter deformation of universal enveloping algebra. Then I explain the classification of finite dimensional representations of U_q(sl_2) and sketch the general case.
09/19
Speaker: Davis Lazowski
Title: RTT presentations, quantum affine algebras and Yangians
Abstract: We’ll explain the RTT presentation of quantum groups and the relation to the Drinfeld presentation. We’ll use this to define quantum affine algebras and Yangians. Time permitting, we will explain the Drinfeld polynomial classification of finite dimensional representations in type A1.
09/26
Speaker: Che Shen
Title: Shifted quantum affine algebras
Abstract: I will review the definition of quantum affine algebra, loop-weight space and the Drinfeld polynomial associated with a highest weight representation. Then I will introduce the notion of shifted quantum affine algebras and give some examples of their representations. Time permitting, I will talk about the category O for shifted quantum affine algebras.
10/03
Speaker: Fan Zhou
Title: Triangular-based algebras
Abstract: [Due to the flu our series on KLR will begin next week; to fill in the time, this week] I will introduce Brundan's theory of triangular-based algebras. Many diagram algebras/categories fit this bill (e.g. Temperley-Lieb, Brauer, Lauda's Uqsl2, you name it), and such algebras have a very nice representation theory, for example admitting families of ``big Vermas'' and ``small Vermas'', in close analogue with the highest weight theory of quasihereditary algebras. In this talk we give an overview of this theory and do some case studies.
10/10
Speaker: Yifan Wu
Title: KLR begins: nil-Hecke
Abstract: I will diagrammatically define the nilHecke algebra, which is the "1-color" case of KLR, and show its algebraic properties. Then I will introduce the categorification theorem for nilHecke Grothendieck groups by Khovanov-Lauda. [NB: cf. Batman Begins]
10/17
Speaker: Igor Chaban
Title: The multicolor KLR
Abstract: The talk will be about general KLR algebra associated to some graph. I will give a definition, describe basis and define a representation in the space of polynomials. Representation theory further will be discussed. In particular, some identities on characters will be obtained. [NB: cf. The Dark Knight]
10/24
Speaker: Fan Zhou
Title: The multicolor KLR categorifies
Abstract: Corresponding to the last movie in the Dark Knight trilogy, this talk will finish up the loose ends in the basic theory so far and venture a little into some later developments. We will state and sketch a proof of the infamous categorification result of KLR; then we will move into cellular territory (type A only). We will briefly sketch what a cellular algebra is, define the cyclotomic KLR algebras, and briefly state the Hu-Mathas cellular basis for the cyclotomic KLR algebra. Then we will briefly sketch what an affine cellular algebra is, and briefly state the Kleshchev-Loubert-Miemietz affine cellular basis for the KLR algebra. Disclaimer: possibly we may sacrifice some statements in favor of sketching proofs, in which case the remaining material will be relegated to a future talk. [NB: cf. The Dark Knight Rises]
10/31
Speaker: Fan Zhou
Title: Cellular structures in (cyclotomic) KLR
Abstract: Last week's plan was a little too ambitious and we only finished the categorification. This week I will define cellular algebras, define the cyclotomic KLR, and construct the cellular basis of the cyclotomic KLR (due to Hu-Mathas). If time permits, I will also extend this to the affine case and construct the affine cellular basis for the entire KLR (due to Kleshchev-Loubert-Miemietz).
11/07
Speaker: Che Shen
Title: Macdonald polynomials and representations of quantum groups
Abstract: I will talk about a result of Etingof and Kirillov in the ‘90s which relates Macdonald polynomials to quantum groups. They considered the intertwiners between certain representations of the quantum group U_q(gl_n) and showed that its trace can be identified with Macdonald polynomials after normalization.
11/14
Speaker: Davis Lazowski
Title: An introduction to Olshanetsky-Perelomov type integrable systems
Abstract: I will explain the basic theory of Olshanetsky-Perelomov-type integrable systems and their relation to rational Cherednik algebras, including basic properties of category O. I will emphasise the type A example, where the integrable system is of Calogero-Moser type and nonsymmetric Jack polynomials appear.
11/21
Speaker: Ivan Danilenko
Title: KLRW algebras
Abstract: We will consider (unflavored) KLRW algebras. I will outline how these generalize KLR algebras, how the braid group acts, and what the applications are.
11/28
Speaker: N/A
Title: Thanksgiving
Abstract: It is Thanksgiving.
12/03
at 4:15 pm in room 528(note the date/time change!)
Speaker: Ivan Danilenko
Title: KLRW algebras, continued
Abstract: This is a continuation of last week's talk.
12/12
Speaker: Elijah Bodish (MIT)
Title: Spin link homology
Abstract: I will explain how folding Khovanov-Rozansky’s SL(2n) homology gives a new approach to categorifying the spin colored SO(2n + 1) Reshetikhin-Turaev link polynomial. To develop this new approach I will mention: skew Howe duality, categorical braid group actions, i-quantum groups, and graphical calculus for SO(2n + 1) centralizer algebras. This is based on joint work with B.Elias and D.Rose (arXiv:2407.00189).
01/21/2025
Speaker: Cailan Li (Academia Sinica)
Title: The boson-fermion correspondence and free fermionic field theory (tentative)
Abstract: TBD

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