Modular representations and cohomology for algebraic, finite and quantum groups

PI将建立在他们最近的工作上的半简单代数群的上同调群的界限。对于大素数，这直接和间接地涉及量子群，在PI开创的方法中。有直接的后果，从一个经典的“通用上同调”理论，获得了PI多年前在合作与其他作者，渐近估计有限群的李型，与更现代的方法有时允许这些被改善到实际的界限。这个程序已经进行了1度上同调，并为所有更高的程度的一般上同调的点。这些结果导致重要的新问题，涉及增长率的上同调空间。同样，结构中的问题是交织在一起的，代数群的研究对于分析量子情况是一个决定性的优势，两种结构都有助于估计Kazhdan-Lusztig多项式的大小。 对于代数群，有许多悬而未决的问题，特别是多项式增长率的问题。这样的增长率问题在数学中广泛存在，特别是在算法问题中，并且在理论计算机科学中很突出。PI还将继续研究有限维代数的Koszul性质，这些代数出现在半单群和量子群的表示中。Koszul结构的产生，或者可能是由几何考虑（反常层及其过滤），但作者一直在推动完全代数方法到几何可能不直接适用的领域。 应用程序的分级和过滤结构的标准（外尔）模块已经发现，与预期的其他结果，以及应用程序的过滤决议和上同调群这些模块。这项工作经常使用一个猜想由于Lusztig，这已被证明是正确的大素数。这些研究展示了这个猜想的更深层次的结果（甚至可以为在更多情况下建立它提供见解）。最后，PI将继续他们在小特征和支持簇的计算方面的工作，该计划涉及重要代数结构的表示和上同调理论，包括半单代数群及其有限和无穷小子群，量子群和Kazhdan-Lusztig多项式。这些结构是相互关联的，因此可以一起进行有益的研究。一个中心方面包括表示的重要类的有限群。在过去的世纪，连续群的类似理论在量子理论和基本粒子理论中发挥了重要作用。事实证明，它们的有限类似物在通信和数据存储设备的设计中很有价值。虽然这个有限理论还很不完善，但它在未来肯定会更加重要。该项目还指出了研究生和本科生多方面参与的未来。

The PIs will build on their recent work on bounds on cohomology groups for semi-simple algebraic groups. For large primes, this both directly and indirectly involves quantum groups, in methods pioneered by the PIs. There are immediate consequences, from a classical "generic cohomology" theory, obtained by the PIs years ago in collaboration with other authors, to asymptotic estimates for bounds for finite groups of Lie type, with more modern methods sometimes allowing these to be improved to actual bounds. This program has been carried out for degree 1 cohomology, and to the point of generic cohomology for all higher degrees. These results lead to important new questions involving the rates of growth of the cohomology spaces. Again the issues in the structures are intertwined, and the study of algebraic groups is a decided advantage for analyzing the quantum case, with both structures contributing to estimates for the growth of sizes of Kazhdan-Lusztig polynomials. For algebraic groups, there are many open questions, especially that of a polynomial rate of growth. Such growth rate issues occur broadly in mathematics, especially in algorithmic issues, and are prominent in theoretical computer science. The PIs will also continue to study Koszul properties for the finite dimensional algebras which come up in the representations of semi-simple groups and quantum groups. Koszul structures arise, or may be conjectured, from geometric considerations (perverse sheaves and their filtrations), but the authors have been pushing entirely algebraic methods into areas where geometry may not directly apply. Applications to the graded and filtered structures of standard (Weyl) modules have already been found, with additional results expected, as well as applications to filtrations of resolutions and cohomology groups of these modules. Often this work uses a conjecture due to Lusztig, which has been proved true for large primes. These studies exhibit deeper consequences of the conjecture (and could even provide insight for establishing it in more cases). Finally, the PIs will continue their work in small characteristic and the calculation of support varieties.This proposal concerns the representation and cohomology theory of important algebraic structures, including semisimple algebraic groups and their finite and infinitesimal subgroups, quantum groups, and Kazhdan-Lusztig polynomials. These structures are interrelated, so can be profitably studied together. A central aspect includes representations of important classes of finite groups. Over the past century, similar theories for continuous groups played a large role in quantum theory and the theory of elementary particles. Their finite analogs have already proved valuable in the design of communications and data storage devices. Though this finite theory remains very incomplete, it will surely be even more important in the future. This project also points to the future in its manifold involvement of graduate and undergraduate students.