群概型、李代数函子和抽象群的Schur-Weyl对偶
2024.06.28
投稿：龚惠英部门：理学院浏览次数：
活动信息
报告题目 (Title)：群概型、李代数函子和抽象群的Schur-Weyl对偶（Schur Weyl Duality for groups schemes, Lie algebra functors, and abstract groups）
报告人 (Speaker)： Zongzhu Lin教授（Kanass State University）
报告时间 (Time)：2024年7月1日 (周一) 9:00
报告地点 (Place)：校本部FJ303
邀请人(Inviter)：张红莲教授
主办部门：理学院数学系
报告摘要：Classically, Schur-Weyl duality was stated between the representations of the symmetric groups Sr over C and the representations of the group GLn(C). It then appeared in many different forms, in terms of Lie algebras gln(C) and also as algebraic group GLn over C. The reason is that the category of finite dimensional representations of these three objects: GLn(C) (as an abstract group), gln as a complex Lie algebra, and GLn as a complex algebraic groups are isomorphic. The question of Schur-Weyl duality was also studied over more general fields. For the abstract group GLn(F) and the algebraic F group GLn, when F is infinite, the duality was studied by Carter-Lusztig. In fact, Carter Lusztig proved that Schur duality holds for GLn as a group scheme over Z. But when F is finite, duality is no longer true in general if F is too small relative to r. However, for duality for Lie algebras over general field or commutative rings, there are not much known. However, the answer over fields of charastistic 0 is obviously clear from Carter-Lusztig’s argument. In this talk, I will briefly review what group functors are. We will also define what Lie algebra functors, and associative algebra functors are. They can be viewed as a presheaf of Lie algebras or associative algebras over an algebraic scheme. Then I will discuss the Schur-Weyl dualities as Lie algebras and as Lie algebra functors, in camparison to the Group schemes and the group of rational points. I will concentrate on type A case only for simplicity. This is a joint work with Stephen Doty.
