Hidden Gradings in Representation Theory

这个项目涉及李代数、有限群和相关对象的表示理论中的一些不同的项目，如Hecke代数、量子群、Khovanov-Lauda-Rouquier(KLR)代数、Brauer代数、W-代数、Ygians和Lie超代数。支持这一提议的统一思想之一是研究在表象理论中许多重要情况下普遍存在的隐藏分级，以及与量子对象分类相关的思想。其中几个项目涉及KLR代数的表示理论，KLR代数被引入来对量子群进行分类。特别是，PI将开启有限型KLR代数的同调代数的研究，并研究由仿射型KLR代数产生的一种新的虚Schur-Weyl对偶现象。这些项目的完成将使我们更好地理解这些特征为零和特征为正的代数。其他项目涉及一般线性李超代数的分次范畴O，Deligne范畴Rep(GL_Delta)，自旋对称群的分次表示，以及有限经典群中极大子群分类的Aschbacher-Scott程序。这项提议有望应用于其他几个数学领域，包括有限群论(及其应用)、李理论、组合学、表示论、纽结理论和范畴理论。表示论是纯数学的核心课题，与数学、数学物理、计算机科学、化学甚至生物学等领域有许多联系。在过去的几年里，这个主题受到了更高范畴理论的严重影响，导致Khovanov，Lauda和Rouquier引入了一些著名的新结构，即KLR代数。这些代数编码了很大一部分组合表示理论背后的更高对称性，包括对称和一般线性群等经典对象。该项目的目标是进一步建立这些代数的理论，并将其应用于提高我们对这些经典对象的理解。该项目的基础研究在未来可能对计算机科学和理论物理产生更广泛的影响。更直接的是，这一奖项将通过对研究生的培训和两个私人投资机构在指导这一领域的其他年轻研究人员方面的持续努力，产生重要的教育影响。该奖项将通过作为主要会议组织者和主要专业期刊编辑的两个个人的积极参与，间接支持在国内和国际上促进这一美丽主题领域的方法和结果的知识。

This project is concerned with some diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Khovanov-Lauda-Rouquier (KLR) algebras, Brauer algebras, W-algebras, Yangians, and Lie superalgebras. One of the unifying ideas underpinning the proposal is the study of hidden gradings ubiquitous in many important situations in representation theory, as well as a related idea of categorification of quantum objects. Several of the projects are concerned with the representation theory of KLR algebras, which were introduced to categorify quantum groups. In particular the PIs will initiate a study of homological algebra of KLR algebras of finite type, and study a new phenomenon of imaginary Schur-Weyl duality arising from KLR algebras of affine type. Completion of these projects will lead to substantially better understanding of these algebras both in characteristic zero and in positive characteristic. Other projects are concerned with graded category O for the general linear Lie superalgebra, Deligne's category Rep(GL_delta), graded representations of spin symmetric groups, and the Aschbacher-Scott program of classifying maximal subgroups in finite classical groups. The proposal is expected to have applications to several other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory. Representation theory is a core topic in pure mathematics, with many connections to other areas of mathematics, mathematical physics, computer science, chemistry and even biology. In the last few years the subject has been influenced heavily by ideas from higher category theory, leading to the introduction by Khovanov, Lauda and Rouquier of some remarkable new structures known as KLR algebras. These algebras encode higher symmetries underlying a large part of combinatorial representation theory, including classical objects like symmetric and general linear groups. The goal of the project is to build further the theory of these algebras and apply it to improve our understanding of these classical objects. The basic research in this project has potential future broader impacts in computer science and theoretical physics. More directly this award will have important educational impact through the training of graduate students and the on-going efforts of both PIs in mentoring other young researchers in this area. The award will indirectly support the promotion of knowledge of the methods and results of this beautiful subject area both nationally and internationally, through the active involvement of both PIs as organizers of major conferences and as editors of leading specialist journals.