Representation Theory, Quantum Groups, and Birational Algebraic Geometry

这个项目致力于研究位于李群、量子群、双曲代数几何和分段线性组合学表示理论的十字路口的区域。基于量子簇代数和几何晶体，提出了一种研究Lusztig正则基和Kashiwara晶基的新方法。从这项研究中得到的新信息将被用于计算约化群的表示的重数和构造新的全正变种。这项研究的结果还将用于解决约化代数群的离散子群表示中出现的问题，以及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论是现代数学中发展最快的领域之一。这一理论对数学的其他领域产生了很大的影响，并在其他自然科学中产生了大量的应用。反过来，正则基和结晶基的概念对于表示理论是非常重要的：仅仅是建立它们的存在就有助于解决经典的计数问题(例如，计算不可约表示的多重性或不可约表示的分解张量积的问题)。因此，任何关于正则碱或晶体碱的信息都将对表象理论非常有益。了解正则基的离散(即组合)和连续(即几何)结构之间的关系是该项目的主要优先事项之一。这种关系已被证明是研究朗兰兹通信的有用工具，朗兰兹通信是20世纪数学中代数和几何之间最神秘和最鼓舞人心的通信。

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.