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

本课题的目的是研究对称群模表示理论中出现的问题。从对称群出发，Schur-Weyl对偶的上同调形式导致人们考虑代数群和Frobenius核的上同调。然后，可以使用代数拓扑学中的技巧来计算产生的一些上同调，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子群和Hecke代数的设置。它们还产生了一些有趣的稳定性结果，这些结果是从计算中出现的，但目前还没有表示论或拓扑解释。它们还导致了第一个已知的具有非零上同调但间隙任意大的模族。上面讨论的同调计算给出了类上同调？关于对称群的Young模的几个定理看似不相关的Frobenius核上同调结果给出了Speht模的“普通上同调”结果。我们将寻求对这些结果的统一解释和推广，例如通过比较Hecke代数在单位根e处的表示理论，因为e是表示场特征的幂。Parshire和Scott证明了著名的Lusztig猜想，在一般线性群的情况下，等价于用对称群模之间的扩张来表示的问题，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，并发现了一些与辫子群上同调有关的特征标理论结果。这里显然还有更多的工作要做。这一建议广泛地属于被称为有限群表示论的数学领域。群是从研究对象的对称性中自然产生的，而对称群是所有群中最自然的。表象理论在物理学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最近取得的进展中发挥了重要作用。表象理论自然而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种程度上了解对称群。许多悬而未决的问题，虽然非常困难，但可以向高级本科生和初级研究生解释。在过去的两年里，该协会曾为代表理论领域的四个高级荣誉论文提供建议。这种及早接触到潜在研究问题的类型可以为未来的博士提供极好的动力。

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.