Combinatorics of Special Functions in Geometry and Representation Theory

主要研究者：海曼，马克 提案编号：DMS -0801262机构：University of California-Berkeley职位：几何和表示理论中特殊函数的组合学海曼教授早期工作的一个主要成果是发现，从2004年开始，麦克唐纳多项式理论中的组合公式，自从麦克唐纳在1988年提出他的多项式以来，（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多项式与最近由组合学家研究的其他特殊q-对称函数，即Lascoux，Leclerc和Thibon的LLT多项式以及Lapointe，Lascoux和莫尔斯的k-舒尔函数联系起来。 从李理论的观点来看，所有这些发展都与一般线性群有关，因此也与A型根系有关。 该研究的指导主题是统一这些最近的组合学发现，将它们与潜在的代数、几何和表示论现象联系起来，并将它们扩展到李群和其他类型的根系。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通道的一部分。 因此，李理论在抽象上就是连续对称性理论。 然而，根据数学中的一个伟大定理，具体的组合数据--根系--支配着最重要的李群的结构。 虽然李群和根系之间的联系是经典的，但还有其他更微妙的与李群理论相关的组合结构，数学家们仍在努力理解。 寻求这种理解的一种方法是开始探索组合方面，这在本质上适合于明确的计算和模式的搜索，然后试图通过引用群论，几何和表示论中更抽象的基本概念来解释观察到的组合现象。 这就是海曼在拟议的研究中寻求追求的理解模式。

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.