Connections between cohomology and representation theory of symmetric groups, braid groups, Hecke algebras, and algebraic groups

建议的项目是研究对称群的模表示理论中出现的问题。从对称群出发，Schur-Weyl对偶的上同调版本导致人们考虑代数群和Frobenius核的上同调。一些出现的上同调可以用代数拓扑学的技巧来计算，特别是迭代循环空间的同调计算。这些计算给出了新的对称群结果，我们期望将其推广到辫群和Hecke代数的情形。他们还产生了一些迷人的稳定性的结果，出现从计算，但目前，没有代表性理论或拓扑解释。他们还导致了第一个已知的家庭模块与非零上同调，但任意大？差距？上面讨论的同源性计算给出？类属上同调对称群的Young模定理。看似无关的Frobenius核上同调结果给？Specht模的“类属上同调”结果我们将寻找一个统一的解释和这些结果的扩展，例如通过比较Hecke代数在e次单位根的表示理论，e是表示域的特征的幂。Parshall和Scott证明了著名的Lusztig猜想，在一般线性群的情况下，等价于一个用对称群模之间的扩张表示的问题，并且与主要研究者的一些对称群结果密切相关。我们将进一步发展这一点。在其他最近的工作中，我们已经开发了组合技术来计算上同调，并发现了一些有关辫子群上同调的特征标理论结果。显然还有更多的工作要做。这一建议福尔斯属于广泛的数学领域被称为代表性理论的有限群。群自然地产生于对物体对称性的研究，而对称群是所有群中最自然的。表示论在物理学和化学中有重要的应用。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。表示理论在许多其他领域自然出现，包括电话网络设计，机器人，分子振动和纠错码。PI认为，这项活动将对高等本科和研究生教育产生更广泛的影响。几乎所有的学生都会在某个时候学习对称群。许多开放的问题，虽然很难，可以解释给高级本科生和研究生开始。在过去的两年里，PI已经为四篇表征理论的高级荣誉论文提供了建议。这种对潜在研究问题的早期接触可以为未来的博士提供很好的动力。

The proposed project is to investigate problems arising in modular representation theory of symmetric groups. Starting from symmetric groups, cohomological versions of Schur-Weyl duality lead one to consider cohomology of algebraic groups and Frobenius kernels. Some of the cohomology that arises can then be computed using techniques from algebraic topology, specifically calculations of homology of iterated loop spaces. These calculations give new symmetric group results, which we expect to extend to the setting of braid groups and Hecke algebras. They also produce some fascinating stability results which emerge from the calculations but, at present, have no representation-theoretic or topological interpretations. They also led to the first known family of modules with nonzero cohomology but arbitrarily large ?gaps?. The homology calculations discussed above give ?generic cohomology? theorems for Young modules of the symmetric group. Seemingly unrelated Frobenius kernel cohomology results give ?generic cohomology" results for Specht modules. We will look for a unified interpretation and extensions of these results, for example by comparing representation theory of Hecke algebras at e-th roots of unity, for e being a power of the characteristic of the representation field. Parshall and Scott proved that the celebrated Lusztig conjecture, in the case of the general linear group, is equivalent to a problem stated in terms of extensions between symmetric group modules, and is closely related to some symmetric group results of the principal investigator. We will develop this further. In other recent work we have developed combinatorial techniques to compute cohomology and discovered some character theory results relating to braid group cohomology. There is clearly much more work to be done here.This proposal falls broadly in the area of mathematics known as representation theory of finite groups. Groups arise naturally from the study of symmetries of objects, and the symmetric group is the most natural of all. Representation theory has important applications in physics and chemistry. In particular, ideas used by mathematical physicists have played an important role in recent progress made in many of the areas described above. Representation theory arises naturally in many other areas, including telephone network design, robotics, molecular vibrations and error correcting codes. The PI believes this activity will have a broader impact on advanced undergraduate and graduate education. Almost all students learn about the symmetric group at some point. Many of the open problems, although very difficult, can be explained to advanced undergraduates and beginning graduate students. In the last two years the PI has advised four senior honors theses in representation theory. This type of early exposure to potential research problems can provide excellent motivation for future Ph.D's.