Representation Theory Seminar
Representation Theory Seminar
Speaker: Taehyeok Heo (Seoul National University)
Date: Thursday, 15 February 2024
Title: The bicrystal structure of the polyhedral realization of $B(\infty)$
Abstract: There are many combinatorial models to describe the crystal $B(\infty)$ or highest weight crystal $B(\lambda)$ for an integral dominant weight $\lambda$. One such model is the polyhedral realization, which is introduced by Nakashima-Zelevinsky. This realization is useful in that it is determined by some linear inequalities and we have an easy combinatorial description of the crystal structure. In this talk, we will explain the polyhedral realization of $B(\infty)$ and $B(\lambda)$. Moreover, we define another crystal structure on $B(\infty)$ and we will show that the set equipped with the two crystal structures we have defined is isomorphic to the (well-known) bicrystal of Kashiwara as bicrystals. If time permits, its generalization to extended crystals will be covered. This is the ongoing joint work with Euiyong Park.
Speaker: Young-Hun Kim (QSMS)
Date: Monday, 22 January 2024
Title: A representation theoretic approach to symmetric functions and quasisymmetric functions
Abstract: This talk will begin by reviewing the well-known connection between the category of modules of symmetric groups and the rings of symmetric functions. We then study an analogous connection between the category $H_\bullet(0)$-mod of modules of 0-Hecke algebras and the ring of quasisymmetric functions. Finally, we will study two noteworthy subcategories of $H_\bullet(0)$-mod, consisting of weak Bruhat interval modules and poset modules, respectively, and discuss the reason why they are interesting subcategories.
Speaker: Jaeseong Oh (Yonsei University)
Date: Monday, 22 January 2024
Title: A tale of q-binomial coefficients and (q,t)-Catalan numbers
Abstract: This presentation aims to illuminate the fascinating interplay between algebraic combinatorics and various branches of mathematics, focusing on two key examples: q-binomial coefficients and (q,t)-Catalan numbers. Initially, I will explain how q-binomial coefficients naturally arise in geometry, specifically within Grassmannians. Subsequently, I will introduce the (q,t)-Catalan numbers and their connection to representation theory, particularly in the context of the Science Fiction Conjecture for Macdonald polynomials. Based on joint work with Donghyun Kim and Seung Jin Lee.