Title: Computation of q-characters
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.
Title: Quiver Hecke algebras and quantum shuffle algebras
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.
Title: Refined canonical stable Grothendieck polynomials and their duals
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.
Title: Explicit formulas for $e$-positivity of chromatic quasisymmetric functions
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.
Title: Sato-Tate distributions and identities of symplectic characters
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.
Title: Visual Studio Code를 사용하여 편리한 LaTeX 환경 구축하기
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.