Derived Equivalences, Braid Relations, and Stability Conditions

表象理论是对对称性和对称性在自然界中的各种表现方式的数学研究。它是代数、几何和组合学的完美结合，它与物理和化学有着卓有成效的相互作用。所提出的研究引入了一种全新的方法来解决表示论中一些基本的未解决的问题，该方法基于最近Broue关于对称群的猜想的证明中发展的导出等价的现代方法。这一方法特别适用于Lusztig关于一般线性群在素性特征下不可约模的特征的著名而有影响的猜想，这一猜想在数学上甚至在表示理论本身之外都激发了重大的进展，并且继续是一个密集的研究主题。研究的第一部分是关于导出等价方法在几个方向上的扩展和应用，包括Broue关于一般线性群在非定义特征下的猜想的证明，最重要的是，在各种Lie型表示理论中，一致地证明了派生范畴上辫子群作用的存在。辫子群作用应该提供一个基础，围绕着它来建立对整个理论家族的更深层次的理解。由于上面提到的关于Broue猜想的工作，Lusztig和James的著名的数值猜想可以用一些小的和可管理的花环积来重新表述。为了探索这种令人惊讶的联系，研究的第二部分利用第一部分中出现的辫子关系以及来自数学物理的一个令人兴奋的新想法，布里奇兰和道格拉斯的稳定性条件，研究了这些花环乘积的同调性质。

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.