Abstract

Jae-Hoon Kwon (Seoul National University)

Title : Introduction to Triangulated Category

Abstract : The notion of triangulated and derived category is now one of the important methods in representation theory of finite-dimensional algebras and Lie algebras. It can be viewed as a generalization of a category of modules of an algebra, which still includes important invariants of the algebra. In this talk, we give an introduction to the triangulated category starting from its definition and then its basic properties. As a main example, we will focus more on the homotopy category of complexes of modules over an algebra. This would be a good motivation to study the language of derived category associated to modules of an algebra. (Reference, A. Zimmermann, Representation Theory; a homological point of view.)

Akito Uruno(Seoul National University)

Title : Representations of symmetric groups

Abstract : In this talk, we will review about irreducible representations of symmetric groups. In detail, I will construct Specht module and prove Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n. Correspondence between representations of the symmetric group and symmetric functions will be breifly explained.

Jeongwu Yu (Seoul National University)

Title : Schubert calculus on Grassmannians..

Abstract : Given any Young Tableaux, it plays a role in the describing the intersection theory orcohomology rings on Grassmannian varieties. In this talk, I will introduce the definition of Schubertvariety and the relationship between Schubert varieties and Young tableaux. Moreover, I will alsointroduce the basic Schubert calculus and Pieri’s formula from the view point of geometry.

Sinmyung Lee (Seoul National University)

Title: Monoidal categories, diagrammatic calculus and categorification

Abstract: Beyond the universal utility of tensor product, its categorical generalization, monoidal category is one of the most fundamental objects in mathematics. A prototype is a category of vector spaces; with the experience in that category, we first review definitions and related notions such as braiding and duality. Next we introduce and practice a diagrammatic calculus in strict monoidal categories. Soon we realize a striking topological nature, which invites us to the world of TQFT and the categorification problem by Crane-Frenkel.

In the latter part we see how the categorification problem has been solved with the tools we have. That is, we present a categorification of the modified quantum group in a diagrammatic way due to Khovanov-Lauda and Rouquier independently. Then it is apparent what 2-representation means, and we give some known constructions. As an application, we obtain a categorification of simple reflections, and hence a derived equivalence of blocks of symmetric groups.

Chul-Hee Lee (Korea Institute for Advanced Study)

Title : Algorithmic exercises in Coxeter groups

Abstract : Coxeter groups play an important role in various areas of mathematics. I will introduce some basic algorithms for Coxeter groups with applications to Lie theory in mind. To understand how these algorithms are applied, I will present many concrete problems, such as finding the length of an element and finding minimal coset representatives.

Hyunse Lee (Seoul National University)

Title : Representation of general linear groups.

Abstract : General linear groups are basic objects in representation theory. In this talk, we will review how one construct irreducible representations of geleral linear groups and classify finite dimensional holomorphic representations. Correspondence between Lie groups and Lie algebras will be included briefly.

Il-Seung Jang (Seoul National University)

Title : The q-character of the representations of quantum affine algebras

Abstract : E. Frenkel and N. Reshetikhin introduced a notion of the “character” to describe the finite-dimensional representations over the quantum affine algebras (of untwisted types), so-called q-character. The q-character can be viewed as a refinement of the usual notion of the character because it works well on the category of finite-dimensional representations over quantum affine algebras rather than the usual character. For example, the q-character induces an injective homomorphism from the Grothendieck ring to the set of Laurent polynomials with the certain variables (related to “l-weights”). Also, the q-characters are related to the universal R-matrix that is extremely important in the representation theory of quantum affine algebras.

In this talk, I first explain the definition of the q-characters. Next, we review some results related to the q-characters. Finally, we consider the combinatorial aspect of the q-characters. In particular, I explain the oriented graphs related to the monomials of the q-character, and then I briefly review a result due to Nakajima in which the oriented graph of the "standard" module has the crystal graph structure.

Seung-Il Choi (Seoul National University)

Title: Schur P-polynomials and crystal bases for the quantum queer superalgebra

Abstract: In this talk, we will explain the connection between Schur P-polynomials and highest weight modules over the queer superalgebra.

To do this we will review the crystal basis theory for the quantum queer superalgebra due to Grantcharov, Jung, Kang, Kashiwara and Kim.

Taehyeok Heo (Seoul National University)

Title : Affine Weyl Group and Hecke Algebra

Abstract : In this talk, we review the notion of affine Weyl group and its Hecke algebra.

Byung-Hak Hwang (Seoul National University)

Title : Kazhdan-Lusztig polynomials and left cell representations

Abstract : In their seminal paper [Kazhdan-Lusztig, Representations of coxeter groups and hecke algebras, Inventiones math., 1979], Kazhdan and Lusztig introduced an important polynomial, so-called the Kazhdan-Lusztig polynomial. While the definition of Kazhdan-Lusztig polynomials comes from the theory of Hecke algebras, they have various connections with other areas. In the first half of this talk, I review some basic facts for Kazhdan-Lusztig polynomials and their connections with geometry and combinatorics. In the second half, I reveiw the left cell representation, which is a W-module constructed via data of Kazhdan-Lusztig polynomials.