SNU Workshop for
Representation Theory and Related Topics

July 27 – 29, 2026

Incheon National University

Speakers

이승빈

서울대학교

이재현

서울대학교

오동건

서울대학교

천지원

서울대학교

최동준

서울대학교

이소연

서울대학교

Sylvain Carpentier

서울대학교

Akito Uruno

NCTS

Organizers

권재훈

서울대학교

서의린

서울대학교

장일승

인천대학교

Venue

Lecture room

Incheon National University

College of Natural Sciences, room 335 (자연과학대학 5호관 335호)

Address: 인천광역시 연수구 아카데미로 119

인천대학교 지도에서 바로찾기

Accommodation

Bridge Hotel Incheon Songdo

Website: http://bridgehotelincheonsongdo.com

Address: 인천광역시 연수구 컨벤시아대로 233

브릿지 호텔 지도에서 바로찾기

Banquet

한마음정육식당 송도타임스페이스점

Address: 인천 연수구 하모니로 158 송도타임스페이스 B동 지하1층

한마음정육식당 지도에서 바로찾기

Supported By

Center for Quantum Structures in Modules and Spaces

Incheon National University

National Research Foundation of Korea

Workshop Poster

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SNU Workshop for Representation Theory 2026 Poster

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Workshop Timetable

Day 1

July 27 (Mon)

14:00 - 14:50
Session 1

이승빈 (서울대학교)
Coffee Break 14:50 - 15:10
15:10 - 16:00
Session 2

이재현 (서울대학교)
Coffee Break 16:00 - 16:20
16:20 - 17:10
Session 3

오동건 (서울대학교)

Day 2

July 28 (Tue)

10:00 - 10:50
Session 4

천지원 (서울대학교)
Coffee Break 10:50 - 11:10
11:10 - 12:00
Session 5

최동준 (서울대학교)
Lunch Break 12:00 - 14:00 (2 Hours)
14:00 - 14:50
Session 6

이소연 (서울대학교)
Coffee Break 14:50 - 15:10
15:10 - 16:00
Session 7

Sylvain Carpentier (서울대학교)
Coffee Break 16:00 - 16:20
16:20 - 17:10
Session 8

Akito Uruno (NCTS)

Day 3

July 29 (Wed)

10:00 - 12:00
Free Discussion

Open discussion on representation theory and related collaborative topics.

Titles & Abstracts

이승빈 | 서울대학교

Title: Diagrammatic Categorification of quantum $\mathfrak{sl}_2$ and Quiver Hecke Algebras

Abstract: Quiver Hecke algebras, also known as Khovanov-Lauda-Rouquier (KLR) algebras, provide a categorification of the negative half of quantum groups. More precisely, the split Grothendieck ring of the category of finitely generated projective modules over a given quiver Hecke algebra is isomorphic to the negative half of the corresponding quantum group. In this talk, we focus on the diagrammatic categorification of the modified quantum $\mathfrak{sl}_2$ via the graphical calculus of a 2-category, to elucidate the underlying structure and origin of the defining relations of these algebras.

이재현 | 서울대학교

Title: Category of Simple Affine Vertex Algebra Modules and Modularity using Characters

Abstract: The simple affine vertex algebra $L_k(\mathfrak{g})$ can be viewed as the simple module $L(k,0)$ of level $k$ over an affine Lie algebra, analogous to $L(0)=\mathbb{C}$ in the finite-dimensional case. While $\mathbb{C}$-Vect possesses a trivial (symmetric) modular tensor category structure, the situation for $L_k(\mathfrak{g})$-Mod is more complicated. In this presentation, first I introduce $L_k(\mathfrak{g})\text{-Mod}_{\text{ord}}$ for specific levels $k$ which has a structure of a semisimple ribbon category with modularity. Second, I provide a systematic method to determine the modularity of $L_k(\mathfrak{g})\text{-Mod}_{\text{ord}}$ by utilizing characters and modular forms. Finally, I demonstrate the extension of this method to the super case by employing supermodularity and supercharacters.

오동건 | 서울대학교

Title: Finite Dimensional Representations of Quantum Affine Algebras and Symmetries of q-Characters

Abstract: Highest weight modules and classical character theory effectively describes finite-dimensional representations of a simple Lie algebra. This talk surveys how a parallel picture appears for quantum affine algebras. Starting with $U_q(\widehat{\mathfrak{sl}}_2)$, we introduce Drinfeld's realization, $\ell$-highest weights, and the classification of finite-dimensional irreducible modules by Drinfeld polynomials. We then explain $q$-characters as refinements of ordinary characters, both from the universal $R$-matrix and from generalized $\ell$-weight eigenspaces. Finally, we discuss how braid group actions, screening operators, and a Weyl group action on a suitable completion characterize the image of the $q$-character map.

천지원 | 서울대학교

Title: A categorical approach to dual canonical basis

Abstract: The dual canonical basis is a distinguished basis of the quantum coordinate ring, which plays an important role in modern representation theory, quantum cluster algebras, and categorification. Despite its significance, explicit computations involving dual canonical basis elements are often difficult; even determining their multiplicative properties can be highly nontrivial. From the viewpoint of Khovanov--Lauda--Rouquier (KLR) categorification, dual canonical basis elements correspond to simple modules, while dual PBW basis elements are categorified by proper standard modules. Moreover, each simple module arises as the head of an appropriate proper standard module. This categorical realization provides a useful way to approach dual canonical basis elements through the structure of KLR modules. In this talk, I will explain this perspective and discuss how the expansion of dual canonical basis elements in the dual PBW basis can be understood from the categorical viewpoint of KLR algebras.

최동준 | 서울대학교

Title: Poisson structures of deformed $W$-algebras and classical $W$-algebras

Abstract: In 1985, Drinfeld and Sokolov introduced a Hamiltonian reduction of affine Lie algebras to construct Poisson structures on the space of differential operators. The resulting algebraic structures, now known as classical $W$-algebras, have been formulated and understood within the framework of Poisson vertex algebras by De Sole, Kac, and Valeri. Parallel to this differential setting, in 1998, Frenkel, Reshetikhin, and Semenov-Tian-Shansky introduced a Hamiltonian reduction of loop groups to construct Poisson structures on the space of difference operators. The resulting algebraic structure was revealed to be a $q \to 1$ limit of the deformed $W$-algebras (defined by Frenkel and Reshetikhin) in type $A$, and is conjectured for other types. In this talk, we provide an explicit algebraic realization of these deformed $W$-algebras in type $A$. Furthermore, by using the deformed Miura transformation, we establish a limit of the deformed $W$-algebras that recovers the Poisson structures of classical $W$-algebras. We conclude by discussing several future directions.

이소연 | 서울대학교

Title: On simple modules of the Iwahori--Hecke algebra of type $A$ at a root of unity

Abstract: The Iwahori--Hecke algebra of type $A$ is a $q$-deformation of the group algebra of the symmetric group, and its representation theory depends significantly on the value of $q$. In this talk, we focus on the simple modules of the Hecke algebra at a primitive $e$-th root of unity. Although these modules are parametrized by $e$-regular partitions, their dimensions remain poorly understood. We first review lower bounds on their dimensions in terms of $e$-restricted tableaux. We then discuss the LLT algorithm, which can be used to compute decomposition numbers. Finally, using specialization functors, we suggest a possible interpretation of the preceding developments in terms of modules of the $0$-Hecke algebra.

Sylvain Carpentier | 서울대학교

Title: Lax Difference Operators and Integrable Lattices.

Abstract: Lax operators provide a unifying framework for describing integrable systems, from differential equations such as Korteweg-de Vries to lattice hierarchies such as Toda. In this talk, we first review scalar and matrix Lax differential operators and their relation to classical affine $W$-algebras. We then explain how scalar Lax difference operators lead to a rich class of integrable lattice hierarchies and discuss recent results on the classification of finite-band Lax hierarchies. The matrix difference case is illustrated through the Pfaff lattice and its relation to the 2D Toda hierarchy. Finally, we emphasize that differential and difference hierarchies are both a manifestation of a deeper structure called Hirota bilinear equations. This perspective connects the tau functions of some integrable systems with representation theory.

Akito Uruno | NCTS

Title: On the Mullineux Conjugation

Abstract: Let $S_d$ be the symmetric group on $n$ letters, and let $D^\lambda$ denote the irreducible $kS_d$-module corresponding to a $p$-restricted partition $\lambda$ of $d$ over a field $k$ of characteristic $p$. In the representation theory of symmetric groups, describing the combinatorial label of the irreducible module obtained by tensoring $D^\lambda$ with the one-dimensional sign representation—denoted as $D^\lambda \otimes \text{sgn}$—has long been a fundamental problem. In 1979, Mullineux introduced an algorithmic bijection on $p$-restricted partitions and conjectured that $D^\lambda \otimes \text{sgn} \cong D^{M(\lambda)}$. In this talk, we review the proof of the Mullineux conjecture following the framework established by Brundan and Kujawa (2003). The core of this approach lies in a superalgebra analogue of Schur-Weyl duality, which connects the representations of $S_d$ to those of the supergroup $GL(n | n)$.

SNU Workshop for Representation Theory and Related Topics

Speakers

이승빈

이재현

오동건

천지원

최동준

이소연

Sylvain Carpentier

Akito Uruno

Organizers

권재훈

서의린

장일승

Venue

Lecture room

Incheon National University

Accommodation

Bridge Hotel Incheon Songdo

Banquet

한마음정육식당 송도타임스페이스점

Supported By

Workshop Poster

Workshop Timetable

Day 1

Session 1

Session 2

Session 3

Day 2

Session 4

Session 5

Session 6

Session 7

Session 8

Day 3

Free Discussion

Titles & Abstracts

이승빈 | 서울대학교

Title: Diagrammatic Categorification of quantum $\mathfrak{sl}_2$ and Quiver Hecke Algebras

이재현 | 서울대학교

Title: Category of Simple Affine Vertex Algebra Modules and Modularity using Characters

오동건 | 서울대학교

Title: Finite Dimensional Representations of Quantum Affine Algebras and Symmetries of q-Characters

천지원 | 서울대학교

Title: A categorical approach to dual canonical basis

최동준 | 서울대학교

Title: Poisson structures of deformed $W$-algebras and classical $W$-algebras

이소연 | 서울대학교

Title: On simple modules of the Iwahori--Hecke algebra of type $A$ at a root of unity

Sylvain Carpentier | 서울대학교

Title: Lax Difference Operators and Integrable Lattices.

Akito Uruno | NCTS

Title: On the Mullineux Conjugation

SNU Workshop for
Representation Theory and Related Topics