Quantum affine algebras and Grassmannians

Abstract：

In this talk, I will talk about the joint work with Wen Chang, Bing Duan, and Chris Fraser on quantum affine algebras of type A and Grassmannian cluster algebras.

Let g=sl_k and U_q(^g) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism Phi from the Grothendieck ring R_l^g of a certain subcategory C_l^g of finite dimensional U_q(^g)-modules to a quotient C[Gr(k,n, \sim)] of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux. Using the isomorphism, we defined ch(T) in C[Gr(k,n, \sim)] for every rectangular semistandard tableau T.

Using the isomorphism and the results of Kang, Kashiwara, Kim, Oh, and Park and the results of Qin, 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 of Arakawa–Suzuki and Lapid-Minguez to the setting of q-characters and obtained an explicit q-character formula for a finite dimensional U_q(^sl_k)-module. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables. We also give a mutation rule for Grassmannian cluster algebras using semi-standard Young tableaux.

About speaker：

Jianrong Li 李建荣

Jianrong Li is a researcher at Faculty of Mathematics of University of Vienna. His research interests include：Mathematics: representation theory, cluster algebras, quantum groups, combinatorics, semigroups. Physics: applications of Grassmannian cluster algebras to scattering amplitudes in physics.