关于q-Schur代数和小q-Schur代数的若干研究

The q-Schur algebras were introduced independently by Dipper-James and Jimbo. These algebras play an important role in the representation and cohomology theories of the finite general linear groups. As homomorphic images of the infinitesimal quantum groups, little q-Schur algebras at odd roots of unity were introduced by Du-Fu-Wang. Later, Fu introduced little q-Schur algebras at the even roots of unity.. This project mainly includes the following detailed contents: Firstly, Doty-Giaquinto and Du-Parshall independently gave an presentation for the q-Schur algebra. In this project, we will give another presentation of the q-Schur algebra. Secondly, we will study the Cartan matrices for little q-Schur algebras. In particular, we will investigate the structures of the irreducible modules and projective modules. Finally, in this project we will give a presentation of little q-Schur algebra at any roots of unity.

q-Schur代数是由Dipper-James和Jimbo分别独立引入的。这类代数在有限一般线性群的表示和同调理论中起着重要作用。作为无穷小量子群的同态像, 奇次单位根下的小q-Schur代数是由Du-Fu-Wang引入的。随后，Fu引入了偶次单位根下的小q-Schur代数。. 本项目主要研究如下内容：1.Doty-Giaquinto和Du-Parshall分别给出了q-Schur代数的表现。本项目将给出q-Schur代数的另一种表现。2.研究奇次单位根下小q-Schur代数的Cartan矩阵。特别地，研究小q-Schur代数的不可约模以及投射模的结构。3.研究任意单位根下小q-Schur代数的表现。

Schur代数及其相关代数是表示论中的重要研究对象。本项目主要研究如下内容：(1) 我们给出q-Schur代数的另一种表现; (2) 研究q-Schur代数的中心; (3) 给出任意单位根下小q-Schur代数的表示型的分类以及奇次单位根下小q-Schur代数u_k(3,3)的表现; (4)研究仿射Schur代数的表现。这些问题有助于q-Schur代数和小q-Schur代数的结构和表示理论的研究。