Sheaf-Theoretic Methods in Modular Representation Theory

模表示理论中的层理论方法

• 批准号: 2202012
• 负责人: Pramod Achar
• 金额: $ 24万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202012&HistoricalAwards=false
• 关键词: Sheaf; Theoretic; Methods; Modular; Representation

A "matrix group" is a set of invertible square matrices that contains all products and inverses of its members. Typical examples include SU(2), the group of 2x2 unitary complex matrices, and O(3,R), the group of orthogonal 3x3 matrices with real entries. Broadly speaking, the subject of representation theory deals with how such groups can act on a complex vector space via linear transformations. One can then ask what happens if we replace the complex numbers by a finite field (or the algebraic closure of a finite field). Modular representation theory is concerned with matrix groups with entries in such a field, acting on vector spaces over the same field. This research will use geometric methods to make advances in modular representation theory. Many of the anticipated results are motivated by known facts in complex representation theory, but new tools and techniques must be developed in the modular case. In connection with this research, the P.I. will also undertake research-training activities aimed at Ph.D. students and other early-career researchers. The past few years have seen the emergence of powerful new tools for applying geometric methods to the representation theory of algebraic groups in positive characteristic, including "parity sheaves" and the "mixed modular derived category." This research will build on these developments with projects on three different topics: (i) the topology of global Schubert varieties; (ii) Kazhdan-Lusztig cells, tensor ideals, and tilting modules; and (iii) "silting" complexes of coherent sheaves. Topic (i) has connections to number theory and to a potential "modular ramified Satake equivalence". Topic (ii) has the most direct links to classical questions in representation theory, while topic (iii) is expected to lead to new avenues of research in K-theory and categorification, for instance in the context of symmetric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

“矩阵组”是一组可逆的平方矩阵，其中包含其成员的所有产品和倒置。 典型的示例包括SU（2），2x2统一复合矩阵和O（3，r）的组，即具有真实条目的正交3x3矩阵。 从广义上讲，代表理论的主题涉及这样的群体如何通过线性变换在复杂的矢量空间上作用。 然后，人们可以询问如果我们用有限字段（或有限场的代数闭合）替换复数会发生什么。 模块化表示理论与在同一字段上作用于矢量空间的矩阵组有关。 这项研究将使用几何方法来取得模块化表示理论的进步。许多预期的结果是由复杂表示理论中的已知事实激励的，但是在模块化案例中必须开发新的工具和技术。 关于这项研究，P.I。还将开展针对博士学位的研究培训活动。学生和其他早期研究人员。 在过去的几年中，出现了强大的新工具，用于将几何方法应用于积极特征的代数群体的表示理论，包括“奇偶脱滑轮”和“混合模块化派生类别”。这项研究将以这些不同主题的项目为基础：（i）全球舒伯特品种的拓扑； （ii）Kazhdan-Lusztig细胞，张量理想和倾斜模块； （iii）相干滑轮的“淤积”复合物。主题（i）与数字理论有联系，并与潜在的“模块化后的萨克克对等”有联系。 主题（ii）与表示理论中的经典问题有最直接的联系，而主题（III）有望导致K理论和分类研究的新途径，例如，在对称空间的背景下。该奖项反映了NSF的法定任务，并通过使用该基金会的知识优点和广泛影响来评估NSF的法定任务，并被认为是值得的。