# QSMS Workshop on Representation Theory

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## Timetable

Week 1

08.09(Mon)

9:30~10:30 장일승

10:45~11:45 박의용

점심시간

14:00~15:00 학생 토론

08.10(Tue)

9:30~10:30 김장수(topic1)

10:45~11:45 이승진

점심시간

14:00~15:00 학생 토론

08.11(Wed)

9:30~10:30 이규환

10:45~11:45 김장수(topic2)

점심시간

14:00~15:00 학생 토론

Week 2

08.17(Tue)

9:30~10:20 우루노아끼또

10:30~11:20 박민희

11:30~12:20 이신명

점심시간

15:00~16:00 학생 토론

08.18(Wed)

9:30~10:20 최동준

10:30~11:20 이현세

점심시간

14:00~15:00 학생 토론

08.19(Thu)

9:30~10:20 허태혁

10:30~11:20 이신명(continued)

점심시간

14:00~15:00 학생 토론

# Title/Abstract(Week1)

Speaker: 장일승

Title: Computation of q-characters

Abstract:

The $q$-character is an important tool to study the finite-dimensional modules over the quantum affine algebra, which was introduced by Frenkel-Reshetikhin. In this talk, I will explain the Frenkel-Mukhin algorithm to compute the $q$-character with several examples.

Speaker: 박의용

Title: Quiver Hecke algebras and quantum shuffle algebras

Abstract:

Let $R$ be a quiver Hecke algebra associated with a generalized Cartan matirx $A$. In this talk, I will explain a connection between finite-dimensional $R$-modules and the quantum shuffle algebra associated with $A$ in the viewpoint of categorification. Using this connection, I will explain that there are certain elements of the upper global basis (dual canonical basis) which can be explained in terms of standard tableaux.

Speaker: 김장수(topic1)

Title: Refined canonical stable Grothendieck polynomials and their duals

Abstact:

In this talk we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi-Trudi-type formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials. We also consider flagged versions of Grothendieck polynomials and their duals with skew shapes. This is joint work with Byung-Hak Hwang, Jihyeug Jang, Jang Soo Kim, Minho Song, And U-Keun Song.

Speaker: 이승진

Title: Explicit formulas for $e$-positivity of chromatic quasisymmetric functions

Abstract:

In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric functions, generalizing chromatic symmetric functions, and conjectured that a chromatic quasisymmetric function of any natural unit interval order is $e$-positive and $e$-unimodal. For a given natural interval order, there is a corresponding partition $\lambda$ and we denote the chromatic quasisymmetric function by $X_\lambda$.

In this talk, I introduce local linear relations for chromatic quasisymmetric functions, and how they can be generalized, which we call Rectangular Lemma. For example, when $\lambda$ is contained in a rectangle such that the width of the rectangle is greater than or equal to the height of the rectangle, one can expand $X_\lambda$ in terms of $X_\mu$'s where $\mu$ is a rectangle, and the lemma states the positive explicit formula for the coefficients of the expansion. Moreover, the similar formula holds for $e$-positivity of chromatic quasisymmetric functions when $\lambda$ is contained in a rectangle. If time permits, we discuss a relation with q-hit numbers, counting certain rook placements. This is a joint work with Sue Kyoung Soh.

Speaker: 이규환

Title: Sato-Tate distributions and identities of symplectic characters

Abstract:

The Sato-Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this talk, we explicitly compute the auto-correlation functions of Sato-Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups, which have interest in their own rights. This is a joint work with Se-jin Oh.

Speaker: 김장수(topic2)

Title: Visual Studio Code를 사용하여 편리한 LaTeX 환경 구축하기

Abstract:

요즘엔 수학을 하는 사람이라면 거의 대부분 LaTeX을 사용합니다. 누구나 쉽게 사용할수 있는 소프트웨어이긴 하지만 LaTeX 작업을 더 효율적으로 하는 방법에 대해 고민한 적이 있으신가요? 본 강연에서는 최근 각광을 받고 있는 Visual Studio Code라는 에디터를 사용하여 LaTeX 작업을 좀더 쉽고 편리하게 하는 방법에 대해 알아봅니다. 본 강연은 아래 링크의 6시간 분량의 동영상 강의를 요약한 것입니다.

https://jangsookim.github.io/lectures/vscode/vscode_lecture0.html

# Title/Abstract(Week2)

Speaker: 우루노아끼또

Title: Crystal bases of quantum groups

Abstract:

In this talk, we will introduce the crystal bases of an integrable module of the quantum group. And next, we will construct the crystal bases of the negative part of the quantum group. The notions of the Demazure module, Weyl group action on the normal crystal will be explained.

Speaker: 박민희

Title: Irreducible highest weight representations of the Virasoro algebra

Abstract:

In this talk, I will briefly introduce irreducible highest weight representations V(c, h) of the Virasoro algebra for all pairs (c, h). I will explain the condition for the representation V(c, h) to be unitary, using the Kac-determinant formula. I will mention how the inequality for the character of V(c, h) can be used to prove the Kac-determinant formula.

Speaker: 이신명

Title: •→•→• vs. •→•←•

Abstract:

The reflection functor defined by Bernstein-Gelfand-Ponomarev has provided a systematic way to relate two quivers with the same underlying graph. Furthermore, it makes computations in representation theory of finite-dimensional algebras, such as Auslander-Reiten quivers, a lot easier. We first study how to make use of the reflection functors to compare module categories of two quivers, when one is obtained from the other by reversing arrows at a source/sink. Then we introduce a tilting functor, a wide generalization of reflection functors and of Morita equivalences.

In the second part, we lift everything to the derived picture and prove that tilting functors give rise to equivalences of derived categories. There are already more tilting modules than Morita progenerators, which hints that derived equivalences should be more interesting than Morita equivalences. If time permits, we give another reason to consider derived categories: tilting theory in a certain orbit category of the derived category suggests how to interpret 'reflections at arbitrary vertex' as mutations of quivers.

Speaker: 최동준

Title: Quantization of Slodowy slices

Abstract:

In this talk, we will see that the Slodowy slice has a natural Poisson structure. And, the Kazhdan grading and filtration will be introduced to construct a quantization of the Slodowy slice.

Speaker: 이현

Title: Skew variants of Robinson-Schensted-Knuth correspondence

Abstract:

Sagan and Stanley introduced variants of Robinson-Schensted-Knuth correspondece for pair of semistandard tableaux of skew shapes. The original motivation was to give a bijective proof of some formulae that Stanley had shown. Recently, Imamura, Mucciconi and Sasamoto showed that such correspondences repect underlying affine crystal structure of type A. In this talk, i will give an introductory review of such correspondences and related identities.

Speaker: 허태혁

Title. Extremal weight modules over quantum infinite rank affine Lie algebras

Abstract:

The notion of extremal weight modules, introduced by Kashiwara, is a generalization of highest weight modules. Even though an extremal weight module does not have a highest weight nor a lowest weight in general, it has several good properties which a highest weight module has.

It is known that two modules are essentially same as $U_q(\mathfrak{g})$-modules, where $\mathfrak{g}$ is of finite type. So many people consider extremal weight $U_q(\mathfrak{g})$-modules when $\mathfrak{g}$ is of affine type. In this talk, I will explain extremal weight $U_q(\mathfrak{g})$-modules when $\mathfrak{g}$ is of (affine) Lie algebras of infinite rank.