学术报告

报 告 人：李建荣 (University of Graz)

报告题目：Quantum affine algebras and Grassmannians

报告时间：2019年10月13日上午10:00-11:00

报告地点：25教14楼数学与统计学院学术报告厅

参加人员：教师、研究生

报告摘要：Let $\mathfrak{g}=\mathfrak{sl}_n$ and $U_q(\widehat{\mathfrak{g}})$ the corresponding quantum affine algebra.

Hernandez and Leclerc proved that there is an isomorphism $\Phi$ from the Grothendieck ring $\mathcal{R}_{\ell}$ of a certain subcategory $\mathcal{C}_{\ell}$ of finite-dimensional $U_q(\widehat{\mathfrak{g}})$-modules to a certain quotient

$\mathbb{C}[{\rm Gr}(n, n+\ell+1, \sim)]$ of a Grassmannian cluster algebra. We proved that this isomorphism induces an isomorphism

$\widetilde{\Phi}$ from the monoid of dominant monomials to the monoid of semi-standard Young tableaux. Using this result and the results of Qin and the results of Kashiwara, Kim, Oh, and Park, we have 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 $T$, where $ch(T)$ is certain map obtained from a formula of Arakawa--Suzuki. We also translated Arakawa--Suzuki's formula to the setting of $q$-characters and apply it to study real modules, prime modules, and compatibility of cluster variables. This is joint work with Wen Chang, Bing Duan, and Chris Fraser.

报告人简介：李建荣，奥地利格拉茨大学博士后。2012年在兰州大学获得博士学位。2013年到2016年在兰州大学数学与统计学院任讲师。曾在以色列希伯来大学，威茨曼科学研究所做博士后。在国际知名期刊 “Int. Math. Res. Not. IMRN”、“Journal of Lie Theory”、“Journal of Algebra”、“J. Algebraic Combin.”、“Algebras and Representation Theory”等上发表论文18篇。主持完成国家自然科学基金青年基金项目1项。