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.
