QSMS Workshop

Speaker : 류미수

Title : Macdonald polynomials and related combinatorics

Abstract : In 1988, Macdonald introduced a family of symmetric functions, called Macdonald polynomials, indexed by partitions and depending on two parameters $q$ and $t$. Ever since the introduction of the polynomials, they have been intensively studied as they can be specialized to most of the previously known symmetric function families. Moreover, Macdonald polynomials arise in various areas of mathematics such as algebraic geometry, representation theory and commutative algebra. In the first part of this talk, we will consider combinatorics related to Macdonald polynomials and the diagonal harmonics. In the second part, we will consider the LLT polynomials and related combinatorics. We will also see why knowing the combinatorics of LLT polynomials is important in studying Macdonald polynomials.

Speaker : 배정

Title : Representations of the symmetric group and the general linear group using tableaux

Abstract : In this talk, we describe some uses of tableaux in studying representations of the symmetric group S_n and the general linear group GL(C). Our main goal is to construct an irreducible representation S^(lambda) of S_n (called a Specht module) and an irreducible representation E^(lambda) of GL(E) (called a Schur or Weyl module) for each partition lambda of n.

Speaker : 권재훈

Title : Introduction to Lie superalgebras

Abstract : In this lecture, we give a brief introduction to Lie superalgebras and their representations. We will cover the following topics:

Lecture 1

Classification of finite-dimensional simple Lie superalgebra

Basic classical Lie superalgebra and their root systems

Lecture 2

Highest weight modules and finite-dimensional irreducible modules (mainly over general linear Lie superalgebras)

The main reference of this lecture is "Dualities and representations of Lie superalgebras" by Cheng and Wang.

Speaker : 이승진

Title : Introduction to chromatic quasisymmetric function

Abstract : Chromatic symmetric functions, certain symmetric functions associated with graphs, are defined by Stanley in 1995 as a generalization of the chromatic number of a graph. Stanley and Stembridge conjectured that if a graph is a incomparability graphs of (3+1)-free poset, the chromatic symmetric function is e-positive. In this talk, we discuss known results of the chromatic (quasi)symmetric such as their Schur positivity and relation with Hessenberg varieties and/or LLT polynomials.

Speaker : 성정현

Title : $e$-positivity of chromatic quasisymmetric functions

Abstract : In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any unit interval graph is $e$-positive. Shareshian and Wachs defined chromatic quasisymmetric functions, which are the quasisymmetric refinement of chromatic symmetric functions, and generalized the conjecture with regard to chromatic quasisymmetric functions. Recently, Abreu and Nigro characterized coefficients of chromatic quasisymmetric functions for the class of unit interval graphs with bipartite complement in the elementary basis as certain $q$-hit numbers.

In this talk, I will explain such characterization with $q$-hit numbers. And then, I will introduce bounce $q$-hit numbers, a refined notion of $q$-hit numbers. We can characterize coefficients of $e_{n-1,1}$ of chromatic quasisymmetric function for any unit interval graph with bounce $q$-hit numbers. This is joint work with Seung Jin Lee.

Speaker : 이소연

Title : Skew Schur polynomials and cyclic sieving phenomenon

Abstract : The cyclic sieving phenomenon was introduced by Reiner, Stanton and White as a generalization of Stembridge's $q=-1$ phenomenon. In this talk, we consider cyclic sieving phenomena associated with skew semistandard Young tableaux, which concern the conjecture proposed by Alexandersson-Pfannerer-Rubey-Uhlin. More precisely, given positive integers $k$, $m$ and a skew partition $\lambda/\mu$, we compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the results thus obtained from the viewpoint of the cyclic sieving phenomenon on semistandard Young skew tableaux of shape $\lambda/\mu$. This is joint work with Y.-T. Oh.

Speaker : 정우석

Title : Towers of algebras and supercharacter theories for combinatorial Hopf algebras

Abstract : It is a fundamental result that the representation theory of the tower of symmetric groups categorifies the Hopf algebra of symmetric functions. In 2006, Bergeron and Li introduced an axiom that guarantees the Grothendieck groups of a tower of algebras form a combinatorial Hopf algebra. Thereby, this axiom serves as a frame for categorifications of combinatorial Hopf algebras. The supercharacter theories were developed to find a more tractable way to understand the representation theory of finite groups, and these provide another frame for categorifications. In this talk, I will review combinatorial Hopf algebras and discuss results on categorifications of combinatorial Hopf algebras using these two frames.

Speaker : 서의린

Title : Basic introduction to an algebraic formalism of integrable systems

Abstract : Abstract: In 1968, Lax discovered a pair of linear operators (Lax pair) which gives rise to all the conservation laws of KdV equation. By an algebraic formalism, Gelfand-Dickii and Adler generalized Lax's arguments and found a series of integrable systems. The algebraic structures induced from the systems are called classical W_N-algebras. In this lecture, we introduce algebraic interpretations of the works of Lax, Gelfand-Dickii and Adler.

Speaker : 송아림

Title : Screening operators for W-algebras

Abstract : In Toda field theory, the W-algebras are defined in terms of the 0th cohomology of BRST complex. In 1992, Frappat, Ragoucy and Sorba found that the W-algebra can also be described as a kernel of the "screening operators". Later, it turned out that one can closely relate the W-algebras with the center of the universal enveloping algebra $U(\hat{\mathfrak{g}})$ of the affine Kac Moody algebra $\hat(\mathfrak{g})$, using this point of view. In this talk, we introduce the definition of W-algebra via BRST complex and explain why it can be described using screening operators. We mainly follow the the paper of Genra(2017).

Speaker : 윤상원

Title : A brief introduction to Wess-Zumino-Witten models

Abstract : Conformal field theory is a field theory with a conformal algebra as a symmetry and the WZW (Wess-Zumino-Witten) model is a nontrivial exactly solvable example of a two-dimensional conformal field theory. Try to understand the WZW-models by understanding the examples such as relationships with Chern-Simons theory (Bulk-boundary correspondence) and the chiral algebra of the models (Sugawara construction, Representations, Fusion rules).

Speaker : 우루노아끼또

Title : Quantum shuffles and q-characters of affine Hecke algebras

Abstract : In this talk, we review how to embed the positive part of a quantum enveloping algebra in a quantum shuffle algebra. Also we review briefly the representation theory of affine Hecke algebras. Finally, when quantum enveloping algebra is of type A, we observe the relation between the image of the dual canonical basis and q-characters of affine Hecke algebras.

Speaker : 이현세

Title : Canonical bases of higher level q-deformed Fock spaces

Abstract : Level $l$ Fock spaces are representations of affine Kac-Moody algebra of A type containing every intergrable highest weight irreducible modules of level $l$. Leclerc and Thibon gave a canonical basis of level 1 Fock spaces in the notion of "IC basis" using bar involution. The basis they gave extends the canonical basis, or global crystal basis of Lusztig and Kashiwara, of the basic subrepresentation of level 1 Fock space. Uglov generalizes this results to higher levels by expressing the transition matrix in terms of parabolic Kazhdan-Lusztig polynomials of affine symmetric groups. In this talk, I will give a brief review on Fock spaces and affine Hecke algebras, and Uglov's construction of canonical basis.

Reference:

D. Uglov, Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, in M. Kashiwara, T. Miwa (Eds.), Physical Combinatorics, in: Progr. Math., vol. 191, Birkhauser, 2000.

B. Leclerc, J.-Y. Thibon, Canonical bases of q-deformed Fock spaces, Internat. Math. Res. Notices 1996, 9, 447-456,

B. Leclerc, Fock space representations, École thématique. Institut Fourier, 2008, pp.40.

Speaker : 허태혁

Title : Crystal structure on King tableaux and oscillating tableaux

Abstract : The character of irreducible finite-dimensional $\mathfrak{sp}_2n$-modules has a well-known combinatorial realization: King tableaux. Whereas the irreducible modules have crystal bases in the sense of Kashiwara's crystals, a crystal structure on King tableaux is not known. Recently, Lee found the crystal structure on King tableaux, which is based on the weight-preserving bijection between King tableaux and oscillating tableaux. In this talk, I will introduce the notion of King tableaux and oscillating tableaux and briefly explain the result of Lee's work. This talk is a summary of Lee's preprint (arxiv:1910.04459, 2020).

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