POSTECH Representation Theory Seminar

General Information: This is a webpage where I keep track of events (sporadically) organized at POSTECH that can roughly be categorified as representation theory (to be understood in the broad, Gelfand sense, to include number theory, geometry, algebraic combinatorics etc.).

Events in the POSTECH-PMI Number Theory Seminar will not be cross-listed here.

Organizer: Valentin Buciumas



Lecture Series on the Gaiotto conjectures

There will be several lectures during the week of February 17-21, 2025. The topic will tentatively be the Gaiotto conjectures and connections to relative Langlands.

Date Time Speaker Affiliation Title Room
February 18 2:00pm-4:30pm Ruotao Yang Academy of Mathematics and Systems Science On the geometrization of the category O of the mixed quantum group: Iwahori fundamental local equivalence Room 404 (math building)
Abstract: D.Gaitsgory introduced the mixed quantum group, whose positive part is the Lusztig quantum group and the negative part is the De Concini-Kac quantum group. It plays an important role in the study of quantum Langlands program. In this talk, we will explain how to obtain an equivalence between its category O and the twisted Whittaker category on the affine flags.
February 20 2:00pm-4:30pm Ruotao Yang Academy of Mathematics and Systems Science On the geometrization of the category of representations of supergroup Room 404 (math building)
Abstract: D. Gaiotto proposed a series of conjectures related to the geometrization of the category of representations of the quantum supergroup. In the limiting case, it says that the category of finite dimensional representations of the (degenerate) supergroup is equivalent to a certain sheaf category on the affine Grassmannian, which can be regarded as a particular case of the local relative Langlands conjecture proposed by Ben-Zvi-Sakellaridis-Venkatesh. In this talk, we will explain how to establish a geometrization of representations of the degenerate supergroup associated with GL(M|N). It is based on a joint work with R. Travkin.
February 21 10:00am-12:30pm Ruotao Yang Academy of Mathematics and Systems Science On the geometrization of the category of representations of quantum supergroup: Gaiotto conjecture Room 404 (math building)
Abstract: D. Gaiotto's conjecture says that the category of finite dimensional representations (resp, category O) of the quantum supergroup is closely related to a certain twisted sheaf category on the affine Grassmannian (resp, affine flags). It is the quantum extension of (a particular case of) the relative Langlands conjecture. In this talk, we will explain recent progress on this conjecture. It is based on ongoing and in-preparation works joint with M.Finkelberg and R. Travkin.




Past events in chronological order



Lecture Series on Macdonald polynomials

Co-organized with Kyoung-Seog Lee . There will be two lectures on October 22 and October 24, 2024. Each lecture will be 2-2.5h long.

Date Time Speaker Affiliation Title Room
October 22, October 24 3:00pm-5:30pm Jaeseong Oh Korean Institute for Advanced Study Introduction to Macdonald polynomials Math Building Room 213
Abstract: Macdonald polynomials form a distinguished basis for the ring of symmetric functions, with profound connections to combinatorics, geometry, representation theory, and probability. Over the past few decades, significant attention has been directed towards eigenoperators for Macdonald polynomials, such as the $\nabla$ operator and the action of the elliptic Hall algebra on symmetric functions. These developments have led to the celebrated $(n+1)^{n-1}$ theorem and the famous Shuffle Theorem, rich with elegant geometry and combinatorics, and have also uncovered fascinating links to knot homologies. In this lecture series, I will try to cover the following topics:
  1. Fundamentals of symmetric functions
  2. The Macdonald polynomials
  3. The Shuffle Theorem and the Science Fiction Conjecture
  4. The elliptic Hall algebra and Khovanov-Rozansky homology of knots




Seminar Day -- November 15

There will be two talks in representation theory on November 15, 2024.

Date Time Speaker Affiliation Title Room
November 15 12:40pm Jae-Ho Lee POSTECH / University of North Florida The universal DAHA of type $(C^\vee_1, C_1)$ and the Askey-Wilson polynomials Math Building Room 104
Abstract: The double affine Hecke algebra (DAHA) for a reduced root system was introduced by Cherednik and later extended by Sahi to include nonreduced root systems. We consider the DAHA $H$ associated with the root system $(C^\vee_1, C_1)$, the most general DAHA of rank one. We introduce a central extension of $H$, called the universal DAHA $\hat{H}_q$ of type $(C^\vee_1, C_1)$. We study the module structure of $\hat{H}_q$ and explore its connection to the Askey-Wilson polynomials, a four-parameter family of $q$-hypergeometric orthogonal polynomials that sit at the top of the Askey scheme hierarchy.
November 15 1:40pm Jonathan Axtell Sungkyunkwan University Quantum generalized Kac–Moody algebras via Hall algebras of complexes Math Building Room 104
Abstract: We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac–Moody algebra into a localized Hall algebra of $\mathbb{Z}_2$-graded complexes of representations of a quiver with (possible) loops. To overcome difficulties resulting from the existence of infinite dimensional projective objects, we consider the category of finitely-presented representations and the category of $\mathbb{Z}_2$-graded complexes of projectives with finite homology.




Lecture Series on the hybrid (mixed) quantum group

There will be two lectures on zoom on November 19 and November 21, 2024. Each lecture will be 2-2.5h long.

Date Time Speaker Affiliation Title Zoom
November 19 6:00pm-8:00pm Quan Situ Université Clermont Auvergne Hybrid quantum group and its representation theory I zoom link I
November 21 6:00pm-8:00pm Quan Situ Université Clermont Auvergne Hybrid quantum group and its representation theory II zoom link II
Abstract: The hybrid (or mixed) quantum group is a quantum algebra with triangular decomposition whose positive part is given by the one of Lusztig’s quantum group, and whose negative part is given by the one of De Concini-Kac quantum group. Its category O can be viewed as a quantum analogue of the BGG category O for semi-simple Lie algebra. In the first talk, I will start by recalling basic results on structures and representations of Lusztig’s quantum group, De Concini-Kac quantum group and small quantum group. Then I will introduce the hybrid quantum group and its relation with the quantum algebras above. I will also introduce its category O and discuss some fundamental structures, e.g. block decomposition, linkage principle, BGG reciprocity and translation functors. In the second talk, I will focus on the principal block and the Steinberg block of this category O. The main result is equivalences from these blocks to coherent-sheaf-theoretic incarnations of the (singular) affine Hecke category. As an application, the principal block is a categorification of the periodic Hecke module, and in particular the (graded) multiplicity of simple module inside Verma module is given by the “generic Kazhdan-Lusztig polynomial”.




Lecture Series on Soergel bimodules

There will be two lectures (tentatively on a topic related to Soergel bimodules) on December 3 and December 5, 2024. Each lecture will be 2-2.5h long.

Date Time Speaker Affiliation Title Room
December 3 3:00pm-5:00pm Hankyung Ko Uppsala University Soergel bimodules and representation theory of reductive groups I Room 404 (math building)
December 5 3:00pm-5:00pm Hankyung Ko Uppsala University Soergel bimodules and representation theory of reductive groups II Room 404 (math building)
Abstract: Many classical problems in Lie theory have combinatorial answers in terms of the Weyl group. Such an answer often comes from and is best understood by looking at a higher structure overlying the Weyl group called the Hecke category, also known as the Soergel bimodules.
This talk is an introduction to Soergel bimodules and an illustration of how it solves problems in representation theory. For the latter, we introduce representation theory of reductive groups in positive characteristic and tell a history where quantum groups make an appearance.