Representation Theory, Quantum Groups, and Birational Algebraic Geometry

该项目致力于研究李群、量子群、两族代数几何和分段线性组合的表示理论的交叉领域。提出了一种基于量子簇代数和几何晶体的研究Lusztig正则基和Kashiwara晶体基的新方法。本研究所得的新信息将应用于计算约化群表示的多重性和构造新的全正变异。本研究的结果还将用于解决在约化代数群的离散子群的表示中出现的问题，以及用于解释和阐述相关的组合和几何结构。李代数和量子群的表示理论是现代数学中最具发展活力的领域之一。这一理论对数学的其他领域产生了巨大的影响，并在其他自然科学领域产生了许多应用。正则基和晶体基的概念对表示理论具有重要意义：仅仅建立它们的存在就有助于解决经典的枚举问题（例如，计算不可约表示的复数或分解不可约表示的张量积的问题）。因此，任何关于规范基或晶体基的信息都将对表征理论非常有益。理解规范基的离散（即组合）和连续（即几何）结构之间的关系是本项目的主要优先事项之一。这种关系已被证明是研究著名的朗兰兹对应的有用工具——朗兰兹对应是20世纪数学中最神秘、最鼓舞人心的代数与几何之间的对应。

This project is devoted to investigation of the area lying at the crossroads of the representation theory of Lie groups, quantum groups, birational algebraic geometry, and piecewise-linear combinatorics. A new approach to the study of Lusztig's canonical bases and Kashiwara'scrystal bases is proposed, based on quantum cluster algebras and geometric crystals. New information resulting from this study will be applied to computing the multiplicities for the representations of reductive groups and for constructing new totally positive varieties. The results of this study will also be used for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups as well as for explication and elaboration of related combinatorial and geometric structures. Representation theory of Lie algebras and quantum groups is one of the most dynamically developing fields of modern Mathematics. This theory has a large impact on other fields of Mathematics and generates numerous applications in other NaturalSciences. In their turn, the concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of their existence has helped in solving classical enumeration problems (e.g., the problem of computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations). Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. Understanding the relationship between the discrete (i.e., combinatorial) and continuous (i.e., geometric) structures of the canonical bases is one of the main priorities of this project. This relationship has proved to be a useful tool in the study of a famous Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.