广州数学大讲坛第一期
第二讲——同济大学付强教授学术报告
题目：Affine quantum Schur--Weyl theory
时间：2024年1月6日（周六）下午16：00——17：00
地点：理学实验楼312
报告人：付强教授
摘要:Quantum Schur--Weyl theory refers to a three-level duality relation. At Level I, it investigates the quantum Schur--Weyl duality between quantum $\frak{gl}_n$ and Hecke algebras of type $A$. This is the quantum version of the well-known Schur--Weyl duality which was beautifully used in H. Weyl's influential book. The key ingredient of the duality is the quantum Schur algebras, whose classical version was introduced by I. Schur over a hundred years ago.
At Level II, it establishes a certain Morita equivalence between quantum Schur algebras and Hecke algebras. The third level of this duality relation is motivated by two simple questions. If an algebra is defined by generators and relations, the realization problem is to reconstruct the algebra as a vector space with hopefully explicit multiplication formulas on elements of a basis; while, if an algebra is defined in term of a vector space such as an endomorphism algebra, it is natural to seek their generators and defining relations.
Beilinson--Lusztig--MacPherson gave a geometric realization of quantum $\frak{gl}_n$ by exploring further properties coming from the quantum Schur--Weyl reciprocity. On the other hand, as endomorphism algebras and as homomorphic images of quantum $\frak{gl}_n$, it is natural to look for presentations for quantum Schur algebras via the presentation of quantum $\frak{gl}_n$. This problem was solved by Doty--Giaquinto and Du--Parshall. Thus, as a particular feature in the type $A$ theory, realizing quantum $\frak{gl}_n$ and presenting quantum Schur algebras form the Level III of this duality relation. In this lecture we will talk about affine quantum Schur--Weyl theory and related topics.
报告人简介：
付强，同济大学数学系教授，博士生导师。研究方向：代数群，量子群及其表示。2004年在华东师范大学获博士学位，曾经于2005年2月到7月和2006年7月访问牛津大学，并多次访问新南威尔士大学。2007年入选同济大学优秀青年教师，2010年入选同济大学英才计划中的青年教学科研骨干计划，2012年入选同济大学英才计划中的攀登高层次人才计划。在Adv math，J. algebra、Pacific J. Math等国际著名杂志发表多篇高水平的研究论文。主持国家自然科学基金优秀青年基金项目、国家自然科学基金面上项目、国家自然科学基金青年基金项目、教育部新世纪人才计划、霍英东基金基础研究基金等多项研究项目。
