Representation Theory, Quantum Groups, and Canonical Bases

表示论、量子群和规范基

• 批准号: 0800247
• 负责人: Arkady Berenstein
• 金额: $ 14.19万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800247&HistoricalAwards=false
• 关键词: Representation; Theory; Quantum; Groups; Canonical

The main theme of this proposal is to investigate the area lying at the crossroads of the representation theory of Lie groups, quantum groups, cluster algebras, and noncommutative algebraic geometry. A new approach to the study of Lusztig's canonical bases and Kashiwara's crystal bases is proposed, based on quantum cluster algebras and geometric crystals as developed in recent papers of the proposer. New information resulting from this study will be applied to computing the multiplicities for the symmetric powers of representations of reductive groups, and constructing new totally positive varieties. The results of this study will be applied for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups and Cherednik algebras as well as for explication and elaboration of related combinatorial and geometric structures including the ``geometric lifting'' of crystal bases as a new tool in understanding the local Langlands correspondence. Representation theory is one of the most dynamically developing fields of modern Mathematics. It has a large impact in other fields of Mathematics and numerous applications in other Natural Sciences. The concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of existence of such bases has helped in solving classical enumeration problems like computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations. A new class of canonical bases discovered by the proposer is expected to settle an old problem of decomposing symmetric powers of representations. Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. It has been revealed in the works of G. Lusztig and the subsequent works of the proposer that there are algebro-geometric counterparts for the purely discrete canonical and crystal bases: totally positive varieties, geometric crystals, and cluster algebras. Understanding the relationship between these structures underlying the canonical bases is one of main priorities of this proposal. This relationship has proved to be a useful tool in the study of Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.

该提案的主题是研究李群、量子群、簇代数和非交换代数几何表示论的十字路口领域。基于提出者最近论文中开发的量子簇代数和几何晶体，提出了一种研究 Lusztig 正则基和 Kashiwara 晶体基的新方法。这项研究产生的新信息将应用于计算还原群表示的对称幂的重数，并构造新的完全正簇。这项研究的结果将应用于解决还原代数群和切雷德尼克代数的离散子群表示中出现的问题，以及相关组合和几何结构的解释和阐述，包括晶体基的“几何提升”作为理解局部朗兰兹对应的新工具。表示论是现代数学发展最活跃的领域之一。它对数学的其他领域和其他自然科学的众多应用都有很大的影响。规范基和晶体基的概念对于表示论非常重要：仅仅建立这些基的存在性就有助于解决经典的枚举问题，例如计算不可约表示的重数或分解不可约表示的张量积。提议者发现的一类新的规范基有望解决分解表示的对称幂的老问题。因此，任何关于规范或晶体基础的信息都对表示论非常有益。 G. Lusztig 的著作和提议者的后续著作揭示了纯离散正则基和晶体基存在代数几何对应物：全正簇、几何晶体和簇代数。理解规范基础背后的这些结构之间的关系是该提案的主要优先事项之一。这种关系已被证明是研究朗兰兹对应关系的有用工具——朗兰兹对应关系是 20 世纪数学中代数和几何之间最神秘和最鼓舞人心的对应关系。