[QSMS Monthly Seminar 2024-10-18] Braid group actions on Grassmannians and extended crystals of type A / Modules of degenerate quantum groups for $P$-partition generating functions

2024년 10월 QSMS  Monthly Seminar

 

• Date:   Oct 18th (Fri) 14:00 ~ 17:00
• Place:  27-220 
• Contents:

 

Speaker:  박의용 (오후2시)

Title:  Braid group actions on Grassmannians and extended crystals of type A

Abstract: Let $\sigma_i$ be the braid actions on infinite Grassmannian cluster algebras induced from Fraser’s braid group actions. Let $T_i$ be the braid group actions on (quantum) Grothendieck rings of Hernandez-Leclerc category $C^0_g$ of affine type $A$, and $R_i$ the braid group actions on the corresponding extended crystals. In this talk, I will talk about three braid group actions $\sigma_i$, $T_i$, and $R_i$ and show that they coincide. This is a joint work with Jian-rong Li.  

 

Speaker: 김영훈 (오후4시)

Title:  Modules of degenerate quantum groups for $P$-partition generating functions
Abstract:  Given a poset $P$ on $\{1, 2, \ldots, n\}$, a $P$-partition is an order-preserving map from $P$ to $\mathbb{Z}_{>0}$. The generating functions of $P$-partitions played a significant role in the development of quasisymmetric functions. The degenerate quantum group $\mathcal{U}_0(\mathfrak{gl}_N)$ is the $q=0$ specialization of the quantum group $\mathcal{U}_q(\mathfrak{gl}_N)$, introduced by Dipper and Donkin. In 1999, Krob and Thibon established a connection between quasisymmetric functions and polynomial $\mathcal{U}_0(\mathfrak{gl}_N)$-modules by defining the characters of these modules. In this talk, we study $P$-partitions and $\mathcal{U}_0(\mathfrak{gl}_N)$-modules. We then construct $\mathcal{U}_0(\mathfrak{gl}_N)$-modules, each of which has a basis consisting of $P$-partitions.