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



Seminar talks on quantum algebras

There will be three talks in the week of November 24-28, 2025. Jianrong Li and Tomasz Przezdziecki will give online talks and Sin-Myung Lee will visit POSTECH on Thursday, November 27.

Date Time Speaker Affiliation Title Room
November 24 6:30pm-9:00pm Tomasz Przezdziecki University of Edinburgh Generalized Schur-Weyl dualities for quantum affine symmetric pairs zoom link
Abstract: In the 90s, Chari and Pressley constructed a Schur-Weyl-type functor linking the representation theories of the affine Hecke algebra of type A, and the quantum affine algebra of sl_n. In 2013, this functor was upgraded and substantially generalized by Kang, Kashiwara and Kim. In their framework, the Hecke algebra is replaced with a graded analogue, the quiver Hecke (or KLR) algebra, and the type of the quantum affine algebra is arbitrary. The construction crucially depends on the properties of meromorphic R-matrices, and is an important tool in, e.g., monoidal categorification of cluster algebras. In my talk, I will explain how to generalize the Kang-Kashiwara-Kim construction to quantum affine symmetric pair coideal subalgebras (also known as affine i-quantum groups). A key new feature is a KLR-type algebra depending on a quiver with an involution. I will also sketch an approach to the representation theory of this algebra based on the categorification of the Enomoto-Kashiwara module and the combinatorics of Lyndon words.
November 26 6:30pm-9:00pm Jian-Rong Li University of Vienna Boundary q-characters of finite-dimensional representations of quantum affine symmetric pairs zoom link
Abstract: Frenkel and Reshetikhin introduced q-characters for finite-dimensional representations of quantum affine algebras, providing a fundamental tool in their representation theory. Together with Tomasz Przezdziecki, we defined boundary q-characters for finite dimensional representations of quantum affine symmetric pairs of split types. In this talk, I will present a new joint work Tomasz Przezdziecki on evaluation modules for split quantum affine symmetric pairs. By computing the action of generators in Lu and Wang’s Drinfeld-type presentation on Gelfand–Tsetlin bases, we determine the spectrum of a large commutative subalgebra arising from this presentation. This leads to an explicit formula for boundary analogues of q-characters, which we interpret combinatorially in terms of semistandard Young tableaux. Our results show that boundary q-characters share familiar features with ordinary q-characters, such as a version of the highest weight property, while also exhibiting new phenomena, including an additional symmetry.
November 27 3:00pm-5:20pm Sin-Myung Lee KIAS Two approaches to finite-dimensional representations of quantum affine superalgebras Room 100
Abstract: In this talk, we discuss two approaches to studying tensor products of finite-dimensional representations of quantum affine superalgebras of type A, which led to monoidal categorifications of cluster algebras in the non-super case. In the first half I will explain how to use the R-matrix to analyze the structure of tensor products of two irreducibles, one of which is real (tensor square is again irreducible). This method is more natural in the perspective of affinizing familiar representation theory of quantum groups (so Lie algebras), and so nicely compatible with the non-super case under our quantum affine super duality (jointly with J.-H. Kwon). In the second half, I will talk about the q-character, the character theory in terms of more standard Drinfeld presentation, and its relation with a q-deformation of the Cartan matrix also recently found in describing q-Y-algebras from quantum toroidal gl_1 (in the context of q-AGT). This gives us a practical combinatorial algorithm to compute q-characters of various representations, which can be very useful both theoretically and computationally.




Past events



Seminar talk on quantum wreath products

Chun-Ju Lai will visit POSTECH September 15-19, 2025 and give a talk on quantum wreath products.

Date Time Speaker Affiliation Title Room
September 16 3:00pm-5:00pm Chun-Ju Lai Academia Sinica Polynomial quantum wreath products Room 404
Abstract: Recently, there have been several work defining various analogs of affine Hecke and Schur algebras, e.g. Kleshchev-Muth (motivated by Turner's conjectures about blocks of symmetric groups), Miemietz-Stroppel (motivated by representations of p-adic $GL_n$), and Song-Wang (who consider diagrammatic presentations from a monoidal category perspective). I will introduce a serious attempt to unify and generalize these many variations, called the quantum wreath products, which afford nice structure and representation theory such as the existence of PBW bases, and Schur-Weyl duality. We will focus on the quantum wreath products of polynomial type and use Vigneras' pro-p Iwahori-Hecke algebras for $GL_n$ as a running example.




Seminar talk on Automorphic Methods in the Geometry of Numbers

Seungki Kim will visit POSTECH May 20-29, 2025 and give a talk on the geometry of numbers.

Date Time Speaker Affiliation Title Room
May 28 2:00pm-4:30pm Seungki Kim University of Cincinnati Automorphic methods in the geometry of numbers Room 404
Abstract: The geometry of numbers concerns itself with metric properties of lattices in a Euclidean space: the length of the shortest nonzero vector of a lattice L, the covering radius of L, upper and lower bounds on these quantities, and so on. Lattices are naturally identified with points of SL(n,Z)\SL(n,R), and lattice-point counting functions are pseudo-Eisenstein series, so the methods of automorphic forms naturally come into play. I'll give a brief introduction to the field, and then discuss my recent work with Seokho Jin (Chung-ang U.) on truncated L^2-norm formula on the number of sublattices of fixed rank and bounded height. Curiously, the L^2-norm diverges in general, so the use of truncation and the Maass-Selberg relations is somewhat forced. As an application, we present a progress on an analogue of the Gauss circle problem for rational points on Grassmannians.




Seminar talk on the Asymmetric Simple Exclusion Process

Travis Scrimshaw will visit POSTECH March 10-12, 2025 and give a talk on ASEP.

Date Time Speaker Affiliation Title Room
March 11 3:00pm-4:00pm Travis Scrimshaw Hokkaido University ASEP on a Ring and Corner Transfer Matrices Room 313
Abstract: ASEP is a “toy” model where fermionic particles move left and right on a discrete lattice. Here we consider the lattice on a circle in continuous time and construct the steady state solution using the Matrix Product Ansatz, which postulates the claim as a trace of a product of matrices. Our solution is given by constructing a corner transfer matrix (in the sense of Baxter) using a quantum oscillator valued 5 vertex model, but unlike in the usual cases, it does not have the usual weight conservation nor satisfy the Yang-Baxter equation. Despite this, it retains enough structure to incorporate the multiline queuing process of Corteel-Mandelshtam-Williams and prove requisite relations. No knowledge will be assumed. This is based on joint work with Atsuo Kuniba and Masato Okado.




Lecture Series on the Gaiotto conjectures

There will be several lectures on the Gaiotto conjectures and connections to relative Langlands during the week of February 17-21, 2025. All lectures take place in room 404 in the math building.
If nothing goes wrong, there should be recordings of the talks available at the end of the lecture series.

Date Time Speaker Affiliation Title Room Recording Recording password
February 18 2:00pm-4:30pm Ruotao Yang Chinese Academy of Sciences On the geometrization of the category O of the mixed quantum group: Iwahori fundamental local equivalence Room 404 zoom recording 1Mjbu?2w
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 Chinese Academy of Sciences On the geometrization of the category of representations of supergroup Room 404 zoom recording 0%Yr5h+*
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 Chinese Academy of Sciences On the geometrization of the category of representations of quantum supergroup: Gaiotto conjecture Room 404 zoom recording b*D8.$PJ
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.




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.




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”.




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 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