Abstract: This talk introduces the theory of Crystal Bases, developed by Kashiwara and Lusztig, which serves as a powerful combinatorial tool for understanding the representation theory of Drinfeld-Jimbo quantum groups associated with complex semi-simple Lie algebras. When the quantum parameter $q$ approaches 0, the algebraic structure of a representation naturally simplifies into a combinatorial framework called a "Crystal Graph". In this talk, we explore the framework of crystal bases and demonstrate how they reduce the tensor product decomposition of reducible representations for ${\mathfrak{g}\mathfrak{l}}_n$ into the combinatorics of Young tableaux. Finally, we briefly introduce the crystal bases for quantum supergroups, along with a short overview of our recent results concerning the general linear and orthosymplectic cases.

 

Organizers: Harry Richman (NCTS), Wille Liu (Academia Sinica)