Affine quantum Schur-Weyl theory

2022.05.06（星期五），10:00-11:00

点：腾讯会议号：499-508-953

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.