Title1: Introduction to representations of quantum affine algebras and its relation to cluster algebras

Title2: Quantum affine algebras and Grassmannian cluster algebras

Title3: Dual canonical basis for C[N] and C[SLk/N^-]

Speaker:  Prof. Jianrong Li ( University of Graz )

Time: Sun,  Nov.15 2020, PM: 20:00-21:30

          Mon, Nov.16 2020, PM: 20:00-21:30

          Tue,  Nov.17 2020, PM: 20:00-21:30

Location:  Zoom Conference

Meeting Number: 774 8737 6626

Password: iTjVP8d

Inviter: Dr. Ming Liu

Abstract 1：

      In the first talk, I will review Chari and Pressley's work on representations of quantum affine algebras, Frenkel, Reshetikhin, Mukhin's work on q-characters of representations of quantum affine algebras, and Hernandez and Leclerc's work on the connection between quantum affine algebras and cluster algebras.

Abstract 2：

      In the second talk, I will talk about connection between representations of quantum affine algebras and Grassmannian cluster algebras. Let g=sln and U_q(^g) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism from the Grothendieck ring of a certain subcategory of finite dimensional U_q(^g)-modules to a certain quotient of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced a bijection between the set of simple modules and the set of semi-standard Young tableaux. Using this result and the results of Qin and the results of Kang, Kashiwara, Kim, Oh, and Park, we proved that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form ch(T)  for some real (resp. prime real) rectangular semi-standard Young tableau T. We translated a formula orginaly due to Arakawa–Suzuki to the setting of q-characters and obtained an explicit q-character formula for a finite-dimensional U_q(^g)-module. This is join work with Chang, Duan, and Fraser.

Abstract 3：

      In the last talk, I will talk about dual canonical basis of C[N] and a certain quotion of C[SLk/N^-]. Denote by N the subgroup of unipotent upper triangular matrices in SLk. We showed that the dual canonical basis of C[N] can be parameterized by semi-standard Young tableaux. Moreover, we gave an explicit formula for every element in the the dual canonical basis. Let N^- be the subgroup of unipotent lower-triangular matrices and let C[SLk/N^-] be the coordinate ring of the base affine space SLk/N^-. We also gave an explicit description of the dual canonical basis of the quotient of C[SLk/N^-] (identifying the leading principal minors with 1).