Algebraic combinatorics of symmetric functions

我的研究领域是代数组合学。这个程序的核心是外尔的特征标公式，这是特征标理论中的一个基本关系。由赫尔曼·外尔在20世纪20年代介绍，它有许多证明，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涉及其他特征的类似公式，如辛舒尔函数或正交舒尔函数，用于其他根系。一个现代的扭曲外尔的特征公式-一个著名的经典代数恒等式-涉及到增加一个额外的参数的身份，从而变形。这些变形版本的外尔的特征公式的证明，以及相关公式的各种根系，涉及对称函数，tableaux，行列式，和交替符号矩阵，并揭示了很多关于这些身份的结构。对结构的新理解建立了一个框架，这反过来又导致了新的结果，这些结果在数论的一个子域中非常重要，它们具有直接和直接的应用。挑战是以有趣和可利用的方式变形Weyl特征公式，根据给定根系的正确表格解释恒等式，并执行组合证明。为了构建这一理论，我需要一个全面的组合框架与解释tableaux和相应的一套算法（如jeu de taquin，格路径建设）和代数技术（如行列式操作）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（这是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）关于上述结果中所涉及的交替符号矩阵的各种统计量的细化/加权枚举（Behrend和Okada向我建议的方向）和4）从文献（Friedburg和Zhang，Brubaker等人）中证明各种假设。我最近的工作引起了一群从不同角度研究类似结果的分析数论家的注意，邀请他们参加研讨会和会议导致了互利的异花授粉，因此我希望他们成为我的组合框架和这些结果的重要受众。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年代末一系列雄心勃勃且富有远见的猜想，激发了人们对数论与其他数学领域（例如调和分析）甚至物理学（例如弦理论和量子场论）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.**