報告題目：Schur-Weyl Duality for groups schemes, Lie algebra functors, and abstract groups

　　報告人：林宗柱（美國堪薩斯州立大學）

　　報告時間：2025.6.30 14：00-15：00

　　報告地點：理學院1-301

　　報告摘要：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.

　　報告人簡介：林宗柱，美國堪薩斯州立大學終身教授，博士生導師，曾任美國科學基金會NSF項目主任和《中國科學：數學》編委。主要從事表示論、代數群以及量子群等方面的研究，論文發表在 Invent. Math.，Adv. Math., Trans. Amer. Math. Soc., CMP 和J. Algebra 等重要學術期刊上，標志性成果包括林-Nakano定理等，是活躍在代數群表示、量子群、Lie代數等研究領域的重要數學家。