Representation Theory

Speaker  : 오재성

Title : Specializing Macdonald Polynomials at t=q^{-k}

Abstract : Macdonald polynomials are a family of symmetric polynomials featuring additional parameters q and t, finding applications across diverse mathematical disciplines. By specializing parameters q and t, one can derive essential symmetric polynomials such as Schur functions (q=t), Hall-Littlewood polynomials (q=0), and Jack polynomials (q=t^α, q→1). In this talk, we explore the specialization at t=q^{-k}. First, we consider the span of Macdonald polynomials of k-admissible partitions and provide a characterization of this subspace due to Feigin--Jimbo--Miwa--Mukhin. Subsequently, we delve into Macdonald polynomial identities at t=q^{-k} implied by the result of Garsia--Haiman--Tesler. Then, we discuss the Schur positivity conjecture for Macdonald polynomials proposed by Haglund--Wilson, along with the related work by Haglund, Yoo, and myself on α -chromatic symmetric functions.

Speaker  : 성정현

Title : An introduction to intersection homology

Abstract : Homology theory is powerful invariant for manifolds which carries important structures: Poincare duality, Morse theory, de Rham isomorphism, hard Lefschetz theorem and etc. Unfortunately, for non-manifolds the structure above breaks down. A new sort of homology, called intersection homology, was introduced by Goresky and MacPherson to study the topology of singular spaces. Intersection homology coincides with ordinary homology for manifolds but it carries above structures for a wide class of singular spaces. In this talk, we define intersection homology and then explain the structures with examples.

Speaker  : 박민희

Title : Introduction to BGG-Verma theorem

Abstract : The most elementary infinite dimensional modules in the category O introduced by Bernstein, Gelfand and Gelfand (abbreviated as BGG in what follows) are Verma modules. Research into Verma modules was originated in Verma's 1966 thesis, in which he gave a sufficient condition for the existence of nontrivial Hom spaces between Verma modules. The necessity of this condition was proved by BGG, who also obtained a new argument of the sufficiency. Another approach to the necessity by Jantzen filtration and contravariant forms is also of independent interest. In this talk, I explain the proof of the well-known BGG-Verma theorem using methods of Jantzen. This is a summary of chapter 5 of the main reference: James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O(2008).

2월 6일 (화)

Speaker  : 남경현

Title : Arithmetic of character variety of reductive groups

Abstract : Counting the number of points on a variety has been a method for investigating its properties, as demonstrated by the Weil conjectures. Nowadays, point counting enables the determination of the E-polynomial via a theorem of Katz. This polynomial provides valuable arithmetic-geometric information about the variety, such as its dimension, number of irreducible components, and Euler characteristic.

In this talk, we will focus on a specific type of variety, the character variety associated to the fundamental group of a surface. This variety plays a central role in various areas of mathematics, including non-abelian Hodge theory, geometric Langlands program, and Yang-Mills theory.  We will discuss this variety when the surface is punctured, with regular semisimple monodromy. A theorem of Frobenius tells us how to compute the number of points of this character variety with the representation theory of finite groups.

Speaker  : 이강산

Title :  BGG resolutions of a finite-dimensional module over a simple Lie algebra

Abstract : A category O introduced by J. Berstein, I. Gelfand, and S. Gelfand in the early 1970s is a full subcategory of U(g)-modules subjected to appropriate finiteness conditions. Let us consider the irreducible highest weight module L(λ) in category O with the highest weight λ such that dim L(λ)<∞. The module L(λ) can be resolved by an exact sequence of direct sum of Verma modules parameterized by equi-length elements of the Weyl group. It is called a BGG resolution of L(λ) and reproduces the Weyl-Kostant character formula for L(λ) as Euler characteristics. In this talk, the construction and application of BGG resolutions will be addressed as a summary of selected topics in chapter 6 of the reference: James E. Humphreys, Representations of Semisimple Lie algebras in the BGG Category O (2008).