Algebraic combinatorics of symmetric functions

我的研究是在代数组合学领域。这个程序的核心是Weyl的特征公式，这是特征理论中的一个基本关系。它由赫尔曼·韦尔在20世纪20年代提出，已经有了许多证明，包括代数和组合。它也有许多推广：即使是经典理论也包括一系列类似的公式，涉及其他特征，如辛舒尔函数或正交舒尔函数，用于其他根系统。韦尔的特征公式--一个著名的经典代数恒等式--的现代扭曲包括在恒等式中增加一个额外的参数，从而使其变形。Weyl特征标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，该框架反过来又导致在数论的一个子领域中重要的新结果，在那里它们有直接和直接的应用。挑战是以有趣和可利用的方式对Weyl特征公式进行变形，根据给定的根系统的正确类型的表解释恒等式，并执行组合证明。*为了构造这个理论，我需要一个全面的组合框架和一套相应的算法(例如Jeu de taquin，格子路径构造)和代数技术(例如行列式处理)来设计和证明这些定理。在具体结果方面，我提出了1)Bn变形Weyl分母恒等式的格路证明(这将是Hamel和King提出的混合代数-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的结果有关)，3)关于先前结果中涉及的交替符号矩阵的各种统计的精化/加权计数(Behrend和Okada向我建议的方向)，以及4)来自文献(Friedburg和Zhang，Brubaker等人)。*我最近的工作引起了分析数论者社区的注意，他们从不同的角度研究类似的结果，并邀请他们参加他们的研讨会和会议，导致了互惠互利的异花授粉，因此我预计他们将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，该计划是20世纪60年代末提出的一系列雄心勃勃而富有远见的猜想，激发了人们对数论与其他数学领域(如调和分析)甚至物理(如弦论和量子场论)之间联系的研究。

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.**