Modular representations and cohomology for algebraic, finite and quantum groups

代数群、有限群和量子群的模表示和上同调

• 批准号: 1001900
• 负责人: Brian Parshall
• 金额: $ 33万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001900&HistoricalAwards=false
• 关键词: Modular; representations; cohomology; algebraic; finite

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.

PI 将建立在他们最近关于半简单代数群的上同调群界限的工作的基础上。对于大素数，这在 PI 开创的方法中直接或间接涉及量子群。从多年前由 PI 与其他作者合作获得的经典“通用上同调”理论，到对李型有限群的边界的渐近估计，以及更现代的方法有时允许将这些改进到实际边界，都会产生直接的后果。该程序已针对 1 阶上同调进行，并达到了所有更高阶上同调的程度。这些结果引出了涉及上同调空间增长率的重要新问题。结构中的问题再次交织在一起，代数群的研究对于分析量子情况具有决定性的优势，这两种结构都有助于估计 Kazhdan-Lusztig 多项式大小的增长。 对于代数群，有许多悬而未决的问题，尤其是多项式增长率的问题。这种增长率问题广泛存在于数学中，尤其是算法问题中，并且在理论计算机科学中尤为突出。 PI 还将继续研究半单群和量子群表示中出现的有限维代数的 Koszul 性质。科祖尔结构是从几何考虑（反常滑轮及其过滤）中产生或推测的，但作者一直在将完全代数方法推向几何可能无法直接应用的领域。 已经发现了对标准（Weyl）模块的分级和过滤结构的应用，预计会产生额外的结果，以及对这些模块的分辨率和上同调群的过滤的应用。这项工作通常使用 Lusztig 的猜想，该猜想已被证明对于大素数是正确的。这些研究展示了该猜想的更深层后果（甚至可以为在更多情况下建立该猜想提供见解）。最后，PI将继续他们在小特征和支持簇计算方面的工作。该提案涉及重要代数结构的表示和上同调理论，包括半简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，因此一起研究可以获益匪浅。一个核心方面包括有限群的重要类别的表示。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重要作用。它们的有限模拟已经被证明在通信和数据存储设备的设计中很有价值。尽管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元化参与的未来。