Modular representations and cohomology for algebraic, finite and quantum groups

PI将建立在他们最近关于半单代数群上同调群的界的工作的基础上。对于大素数，在PI开创的方法中，这直接和间接涉及量子群。从几年前由PI与其他作者合作获得的经典的“类属上同调”理论，到有限李型群的界的渐近估计，都有直接的结果，更现代的方法有时允许将这些改进到实际的界。这个计划已经针对1度上同调进行了，并达到了所有更高程度的类属上同调的程度。这些结果引出了涉及上同调空间的增长率的重要的新问题。同样，结构中的问题是相互交织的，代数群的研究是分析量子情况的决定性优势，这两种结构都有助于估计Kazhdan-Lusztig多项式的大小增长。对于代数群，有许多悬而未决的问题，特别是关于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，在理论计算机科学中也是突出的。PI还将继续研究有限维代数的Koszul性质，这些性质出现在半单群和量子群的表示中。Koszul结构的产生，或可能是猜想，从几何的考虑(倒置的鞘和它们的过滤)，但作者一直在推动完全代数方法的领域，几何可能不会直接应用。标准(Weyl)模的分次结构和滤子结构的应用已经被发现，并期望得到更多的结果，以及应用于这些模的分解和上同调群的过滤。这项工作经常使用Lusztig的猜想，这一猜想已被证明对大素数是正确的。这些研究展示了这一猜想的更深层次的后果(甚至可以为在更多情况下建立这种猜想提供洞察力)。最后，PI将继续他们在小特征和支撑变量计算方面的工作。这项建议涉及到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，因此可以一起研究，这是有益的。一个中心方面包括重要的有限群类的表示。在过去的一个世纪里，连续群的类似理论在量子理论和基本粒子理论中发挥了很大的作用。它们的有限类比已经被证明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，但它在未来肯定会变得更加重要。这个项目还指出了未来研究生和本科生的多方面参与。

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.