Derived Equivalences, Braid Relations, and Stability Conditions

表示论是对对称性和对称性在自然界中表现的各种方式的数学研究。一个奇妙的融合代数，几何和组合学，它享有丰富的互动与物理和chemical.The拟议的研究介绍了一个全新的方法来解决一些基本的未解决的问题表示论，基于现代方法的衍生等价在最近的证明Broue的猜想对称群。这种方法特别适用于Lusztig关于一般线性群的不可约模的特征标的著名的和有影响力的猜想，该猜想激发了数学的重大进展，甚至在表示论本身之外，并且仍然是一个深入研究的主题。包括证明Broue的猜想一般线性群在非定义的特点，最重要的是，一个统一的证明存在的辫子群行动的衍生类别在各种李型表示理论。辫子群作用应该提供一个基础，围绕它建立一个更深入的理解整个家庭的theors.Thanks工作Broue的猜想上面提到的，著名的数值计算Lusztig和詹姆斯可以重新制定的一些小的和可管理的花圈产品。为了利用这一惊人的联系，研究的第二部分调查这些花环产品的同调属性，使用辫子关系出现在第一部分连同一个令人兴奋的新想法来自数学物理，稳定条件的Bridgeland和道格拉斯。

Representation theory is the mathematical study of symmetry and of the various ways symmetry manifests itself in nature. A wonderful blend of algebra, geometry, and combinatorics, it enjoys fruitful interactions with physics and chemistry.The proposed research introduces a completely new approach to some fundamental unsolved problems in representation theory, based on modern methods of derived equivalences developed in the recent proof of Broue's conjecture for symmetric groups. This approach applies in particular to Lusztig's famous and influential conjecture on characters of irreducible modules for general linear groups in prime characterisctic, which has inspired major advances in mathematics even outside representation theory proper and continues to be a subject of intense study.The first part of the research concerns extensions and applications of the derived equivalence methods in several directions, including a proof of Broue's conjecture for general linear groups in nondefining characteristic and, most significantly, a uniform proof of the existence of braid group actions on derived categories in a variety of Lie-type representation theories. The braid group actions should provide a foundation around which to build a deeper understanding of the whole family of theories.Thanks to the work on Broue's conjecture mentioned above, the famous numerical conjectures of Lusztig and James can be reformulated in terms of some small and manageable wreath products. In order to exploit this surprising connection, the second part of the research investigates homological properties of these wreath products, using the braid relations appearing in the first part together with an exciting new idea coming from mathematical physics, the stability conditions of Bridgeland and Douglas.