q-Schur超代数  

q-Schur superalgebras

作　　者：杜杰 顾海霞 Jie Du;Haixia Gu

机构地区：[1]School of Mathematics and Statistics, University of New South Wales (UNSW) [2]湖州师范学院理学院数学系,湖州3130001

出　　处：《中国科学：数学》2018年第11期1571-1594,共24页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11501197和11671234)资助项目

摘　　要：本文总结了近期在q-Schur超代数、量子一般线性超群和它们的典范基以及不可约(多项式)表示方面的研究.首先给出了q-Schur超代数在三种不同背景下的定义和相应的基,并且刻画了这三组基之间的关系,接着描述了q-Schur超代数中的某些乘法公式及其在量子一般线性超群的新实现、q-Schur超代数的正则表示和量子一般线性超群的正部分的典范基的构造中的应用,同时给出了q-Schur超代数的半单性的判别条件.通过对Alperin的权猜想和Scott的置换表示理论的推广,本文得到了q-Schur超代数的不可约模分类.本文最后提到了在不引入量子坐标代数情形下构造无穷小和小q-Schur超代数的新方法.This paper provides a survey on the latest developments of q-Schur superalgebras, quantum linear supergroups and their canonical bases and irreducible（polynomial） representations. We first introduce the definitions of q-Schur superalgebras in three different contexts and their associated bases, and describe the relationships between the three bases. We then move on to display certain multiplication formulas in q-Schur superalgebras and discuss their applications to the new realization of quantum linear supergroups and to the regular representation of a q-Schur superalgebra. We provide another application for the construction of canonical bases for the positive part of the quantum linear supergroup. As a by-product, a semi-simplicity criterion is given for q-Schur superalgebras. Moreover, by generalising the idea of Alperin＇s weight conjecture and Scott＇s permutation representation theory, we provide a classification of irreducible modules for a q-Schur superalgebra. We also mention a new approach to introduce infinitesimal and little q-Schur superalgebras without using quantum coordinate superalgebras.

关 键 词：q-Schur超代数 HECKE代数 量子一般线性超群 

分 类 号：O152.5[理学—数学]