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.

摘要 主要研究员：Haiman, Mark 提案编号：DMS - 0801262机构：加州大学伯克利分校 标题：几何和表示理论中特殊函数的组合 海曼教授早期工作的一个主要成果是从 2004 年开始发现了理论中的组合公式麦克唐纳多项式，自从麦克唐纳提出他的多项式以来，人们一直在寻找这个东西1988 年多项式（Haiman 的这方面研究是与 Jim Haglund 和 Nick Loehr 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 q 对称函数联系起来，即 Lascoux、Leclerc 和 Thibon 的 LLT 多项式，以及 Lapointe、Lascoux 和 Morse 的 k-Schur 函数。 从李理论的角度来看，所有这些发展都与一般线性群有关，因此与 A 型根系统有关。拟议研究的指导主题将是统一这些最近的组合发现，将它们与基础代数联系起来、几何和表示理论现象，并将它们扩展到其他类型的李群和根系统。从更广泛的角度来看，组合数学是处理从抽象到具体的过渡的数学部分。 因此抽象的李理论是连续对称性理论。 然而，根据数学中最伟大的定理之一，具体的组合数据——根系统——控制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与李理论相关的组合结构，数学家们仍在努力理解这些结构。 寻求这种理解的一种方法是从探索组合方面开始，组合方面本质上适合显式计算和模式搜索，然后尝试通过引用群论中更抽象的基本概念来解释观察到的组合现象，几何和表示理论。 这就是海曼在拟议研究中寻求追求的理解模式。