Cohomology and Representation Theory

主要研究对称群及其相关对象的模表示理论和上同调问题，包括代数群、Frobenius核、Schur代数和超代数、Iwahori-Hecke代数。该项目包括使用这些不同对象的表示理论之间的联系来获得新的结果。通过Schur等人的经典工作，对称群的普通表示理论与一般线性群的普通表示理论密切相关。最近，这种关系被扩展到模表示理论，甚至最近扩展到相应的上同调。虽然这种相互作用可以在任何方向上导致新的结果，但它在研究对称群时特别有用，因为许多基本问题仍然是完全开放的。研究者将继续完善和改进技术，以利用这一重要关系，以便在这两个领域获得结果。最近Parshall和Scott证明了著名的Lusztig猜想，在一般线性群的情况下，完全等价于对称群表示理论领域的问题。与此同时，Brundan和Kleshchev、Ariki、Grojnowski等人的突破导致了对称群论的“lie - theory”方法，这与Gordon James等人的原始方法有很大不同。总之，这些发展表明，未来几年在这一领域将是非常令人兴奋的。这个项目是在一个被称为表示理论的数学领域，特别是有限群的表示理论。涉及群体及其表示的问题自然出现在许多不同的领域，包括数学物理、化学、纠错码、密码学等。特别是，数学物理学家使用的思想在上述许多领域的最新进展中发挥了重要作用。

The principal investigator will explore problems in the modular representation theory and cohomology of the symmetric group and related objects, including algebraic groups, Frobenius kernels, Schur algebras and superalgebras, and Iwahori-Hecke algebras. The project includes using connections between the representation theories of these different objects to obtain new results. The ordinary representation theory of the symmetric group is closely related to that of the general linear group through classical work of Schur and others. Recently this relationship has been extended to the modular representation theory, and even more recently to the corresponding cohomology. Although this interplay can lead to new results in either direction, it is especially useful in studying the symmetric group, where many of the basic problems remain completely open. The investigator will continue to refine and improve techniques to exploit this important relationship in order to obtain results in both areas. Recently Parshall and Scott proved that the celebrated Lusztig conjecture, in the case of the general linear group, is equivalent to a problem entirely in the realm of symmetric group representation theory. Meanwhile breakthroughs by Brundan and Kleshchev, Ariki, Grojnowski and others have led to a ``Lie-theoretic" approach to the symmetric group theory which is very different from the original approach of Gordon James and others. Together these developments suggest the next few years will be very exciting indeed in this field.This project is in an area of mathematics known as representation theory, in particular the representation theory of finite groups. Problems involving groups and their representations arises naturally in many diverse fields, including mathematical physics, chemistry, error correcting codes, cryptography and others. In particular, ideas used by mathematical physicists have played an important role in recent progress made in many of the areas described above.