Combinatorics of Special Functions in Geometry and Representation Theory

几何与表示论中特殊函数的组合

• 批准号: 0801262
• 负责人: Mark Haiman
• 金额: $ 54万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801262&HistoricalAwards=false
• 关键词: Combinatorics; Special; Functions; Geometry; Representation

ABSTRACTPrincipal Investigator: Haiman, Mark Proposal Number: DMS - 0801262Institution: University of California-BerkeleyTitle: Combinatorics of Special Functions in Geometry and Representation TheoryA major result of Professor Haiman's earlier work was the discovery, starting in 2004, of combinatorial formulas in the theory of Macdonald polynomials, something that had been sought ever since Macdonald introduced his polynomials in 1988 (this aspect of Haiman's research was carried out in collaboration with Jim Haglund and Nick Loehr). The formulas connect Macdonald polynomials with other special q-symmetric functions recently studied by combinatorialists, namely the LLT polynomials of Lascoux, Leclerc and Thibon, and the k-Schur functions of Lapointe, Lascoux and Morse. From the point of view of Lie theory, all these developments are connected with general linear groups and therefore with the root systems of type A. The guiding themes of the proposed research will be to unify these recent combinatorial discoveries, to connect them with underlying algebraic, geometric and representation theoretic phenomena, and to extend them to Lie groups and root systems of other types.In a broader optic, combinatorics is the part of mathematics that deals with the passage from the abstract to the concrete. Thus Lie theory in the abstract is the theory of continuous symmetries. However, by one of the great theorems in mathematics, concrete combinatorial data--the root systems--govern the structure of the most important Lie groups. While the link between Lie groups and root systems is classical, there are also other, more subtle, combinatorial structures associated with Lie theory, which mathematicians are still striving to understand. One way to seek such understanding is to begin by exploring the combinatorial side, which by nature lends itself to explicit computation and the search for patterns, and afterwards to try to explain the observed combinatorial phenomena by reference to more abstract underlying concepts from group theory, geometry and representation theory. This is the mode of understanding which Haiman seeks to pursue in the proposed research.

项目编号：DMS - 0801262机构：加州大学伯克利分校海曼教授早期工作的一个主要成果是从2004年开始发现麦克唐纳多项式理论中的组合公式，这是自1988年麦克唐纳引入多项式以来一直在寻找的东西（海曼的这方面研究是与吉姆·哈格伦德和尼克·洛尔合作进行的）。这些公式将Macdonald多项式与组合学家最近研究的其他特殊的q对称函数，即Lascoux、Leclerc和Thibon的LLT多项式，以及Lapointe、Lascoux和Morse的k-Schur函数联系起来。从李论的角度来看，所有这些发展都与一般线性群和a型根系统有关。建议研究的指导主题将是统一这些最近的组合发现，将它们与潜在的代数、几何和表示理论现象联系起来，并将它们扩展到李群和其他类型的根系统。从更广泛的角度来看，组合学是数学中处理从抽象到具体的过渡的部分。因此，李论在抽象意义上是连续对称的理论。然而，根据数学中一个伟大的定理，具体的组合数据——根系统——支配着最重要的李群的结构。虽然李群和根系统之间的联系是经典的，但还有其他更微妙的组合结构与李论有关，数学家们仍在努力理解。寻求这种理解的一种方法是从探索组合的一面开始，它本身就适合显式计算和寻找模式，然后尝试通过参考群论、几何和表示理论中更抽象的潜在概念来解释观察到的组合现象。这是海曼在提议的研究中寻求追求的理解模式。