Algebraic combinatorics of symmetric functions

对称函数的代数组合

• 批准号: RGPIN-2015-06126
• 负责人: Hamel, Angele
• 金额: $ 0.8万
• 依托单位: Wilfrid Laurier University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652901
• 关键词: Algebraic; combinatorics; symmetric; functions

My research is in the area of algebraic combinatorics. The heart of this program is Weyl's character formula, a fundamental relation in character theory. Introduced by Hermann Weyl in the 1920s, it has had many proofs, both algebraic and combinatorial. It has also had many generalizations: even the classical theory includes an array of similar formulae involving other characters, such as symplectic Schur functions or orthogonal Schur functions, for other root systems. A modern twist on Weyl's character formula--a well-known classical algebraic identity--involves adding an extra parameter to the identity, thus deforming it. The proofs of these deformed versions of Weyl's character formula, and proofs of related formulae for various root systems, involve symmetric functions, tableaux, determinants, and alternating sign matrices, and reveal much about the structure of these identities. New understanding of the structure builds a framework which in turn leads to new results that are important in a subdomain of number theory where they have immediate and direct application. The challenge is to deform the Weyl character formulae in interesting and exploitable ways, interpret the identities in terms of the right sort of tableaux for the given root system, and execute combinatorial proofs.***In order to construct this theory, I need a comprehensive combinatorial framework with an interpretation in terms of tableaux and a corresponding set of algorithms (e.g. jeu de taquin, lattice path constructions) and algebraic techniques (e.g. determinantal manipulation) to devise and prove these theorems. In terms of specific results I propose 1) a lattice path proof of a Bn deformed Weyl denominator identity (this would be a fully combinatorial proof of a hybrid algebro-combinatorial proof due to Hamel and King), 2) proofs of both the Cn and Dn deformed Weyl denominator identities (related to results of Brubaker and Schultz), 3) refined/weighted enumerations with respect to various statistics of the alternating sign matrices involved in the preceding results (a direction suggested to me by both Behrend and Okada) and 4) proofs of various conjectures from the literature (Friedburg and Zhang, Brubaker et al.).***My recent work caught the attention of a community of analytic number theorists working on similar results from a different angle, and invitations to their workshops and conferences has resulted in a mutually beneficial cross pollination, hence I expect them to be an important audience for my combinatorial framework and these results. The work also has connections to the celebrated Langlands' program in number theory, an ambitious and visionary set of conjectures from the late 1960s that has inspired investigation into connections between number theory and other areas of mathematics (e.g. harmonic analysis) and even physics, e.g. string theory and quantum field theory.**

我的研究方向是代数组合学。这个节目的核心是Weyl的性格公式，这是性格理论中的一个基本关系。它在20世纪20年代由赫尔曼·魏尔（Hermann Weyl）提出，有许多代数和组合的证明。它也有许多推广：即使是经典理论也包括一系列涉及其他特征的类似公式，例如其他根系统的辛舒尔函数或正交舒尔函数。Weyl的特征公式——一个著名的经典代数恒等式——有一个现代的转折，就是给这个恒等式添加一个额外的参数，从而使它变形。Weyl特征公式的这些变形版本的证明，以及各种根系统的相关公式的证明，涉及对称函数、表、行列式和交替符号矩阵，并揭示了这些恒等式的许多结构。对结构的新理解建立了一个框架，这个框架反过来又导致了新的结果，这些结果在数论的子领域中很重要，在那里它们有直接和直接的应用。挑战在于以有趣和可利用的方式变形Weyl字符公式，根据给定根系统的正确表格类型解释身份，并执行组合证明。为了构建这个理论，我需要一个综合的组合框架，用tableaux和相应的一组算法（例如jeu de taquin，晶格路径结构）和代数技术（例如行列式操作）来解释和证明这些定理。就具体结果而言，我提出1)Bn变形Weyl分母恒等式的晶格路径证明（这将是由于Hamel和King的混合代数-组合证明的完全组合证明），2)Cn和Dn变形Weyl分母恒等式的证明（与Brubaker和Schultz的结果相关），3)关于上述结果中涉及的交替符号矩阵的各种统计的改进/加权枚举（Behrend和Okada向我建议的方向）和4)文献中各种猜想的证明（Friedburg和Zhang， Brubaker等人）。***我最近的工作引起了一群分析数论学家的注意，他们从不同的角度研究类似的结果，邀请他们参加他们的研讨会和会议导致了互利的交叉授粉，因此我希望他们成为我的组合框架和这些结果的重要听众。这项工作还与著名的朗兰兹数论计划有关，朗兰兹数论计划是20世纪60年代末提出的一系列雄心勃勃、富有远见的猜想，它激发了人们对数论与其他数学领域（如谐波分析）甚至物理学领域（如弦论和量子场论）之间联系的研究