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

提出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，利用Schur-Weyl对偶的上同调版本，考虑代数群和Frobenius核的上同调。出现的一些上同调可以使用代数拓扑的技术来计算，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，我们期望将这些结果推广到辫群和Hecke代数的设置中。它们也从计算中产生了一些令人着迷的稳定性结果，但目前还没有表示理论或拓扑解释。它们还导致了已知的第一个具有非零上同调但任意大间隙的模族。上面讨论的同调计算给出？通用的上同调吗?对称群的Young模的定理。看似无关的Frobenius核上同调结果给出？Specht模块的“泛型上同源”结果。我们将寻找这些结果的统一解释和扩展，例如，通过比较赫克代数在单位的e次根上的表示理论，因为e是表示域特征的幂。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展表述的问题，并且与一些主要研究者的对称群结果密切相关。我们将进一步开发它。在最近的其他工作中，我们发展了计算上同调的组合技术，并发现了一些与辫群上同调有关的特征理论结果。显然，在这方面还有很多工作要做。这个建议大致属于数学领域，即有限群的表示理论。群从研究物体的对称性中自然产生，而对称群是最自然的。表征理论在物理和化学中有着重要的应用。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。表征理论自然地出现在许多其他领域，包括电话网络设计、机器人、分子振动和纠错码。PI相信这一活动将对高等本科和研究生教育产生更广泛的影响。几乎所有的学生都学过对称群。许多开放性的问题，虽然很难，但可以解释给高年级的本科生和刚开始的研究生。在过去的两年里，PI指导了四篇关于表征理论的高级荣誉论文。这种对潜在研究问题的早期接触可以为未来的博士学位提供极好的动力。

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.