9I果冻制作厂

　　报告题目：Schur-Weyl Duality for groups schemes, Lie algebra functors, and abstract groups

　　报告人：林宗柱（美国堪萨斯州立大学）

　　报告时间：2025.6.30 14：00-15：00

　　报告地点：理学院1-301

　　报告摘要：Classically, Schur-Weyl duality was stated between the represen tations 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 ab stract 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 groups GLn, when F is infinite, the duality was studied by Carter-Lusztig. In fact, Carter and 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 fields or commutative rings, there are not much known although the answer over fields of characteristic 0 is obvious 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 groups of rational points. I will concentrate on the type A case only for simplicity.

　　报告人介绍：林宗柱，美国堪萨斯州立大学终身教授，博士生导师，曾任美国科学基金会NSF项目主任和《中国科学：数学》编委。主要从事表示论、代数群以及量子群等方面的研究，论文发表在 Invent. Math.，Adv. Math., Trans. Amer. Math. Soc., CMP 和J. Algebra 等重要学术期刊上，标志性成果包括林-Nakano定理等，是活跃在代数群表示、量子群、Lie代数等研究领域的重要数学家。