Algebraic, Combinatorial, and Analytic Applications of Symmetric Functions

从晶体的晶格结构，到物质的状态，到矩阵和微分算子，系统的特征及其演化都是通过对称性来分类的。代数组合学研究对称性通过他们的表现在著名的离散对象，如图，排列和分区。它的方法已经成功地解决了其他科学中的问题，如物理学，计算机科学和生物学。这个项目涉及代数组合学的应用，特别是它的子领域对称函数理论，以解决这些问题。这个项目是围绕所使用的工具，即对称函数理论和相关的组合。表示论中的各种复杂性问题涉及计算某些结构常数和多重性，这些结构常数和多重性可以通过对称群的克罗内克系数和体积系数来表达，这些系数可以使用舒尔函数来定义。在统计力学中，菱形镶嵌等可积格点模型的配分函数往往是李群特征，其渐近研究揭示了边界附近的高斯么正系综特征值分布或极限形状和曲面的存在等概率行为。该项目旨在扩展这些应用程序，以研究其他模型和分布。研究作为旗簇上同调类代表的舒伯特多项式的组合和代数性质，可以得到相应结构常数的组合解释。此外，它们的稳定版本，斯坦利对称函数的计算特性，可以导致理解随机排序网络的神秘极限行为，对应于排列到相邻转置的简化分解。

From the lattice structure of crystals, to states of matter, to matrices and differential operators, the traits of systems and their evolution are classified by symmetries. Algebraic combinatorics studies symmetries via their manifestations in well-known discrete objects like graphs, permutations, and partitions. Its methods have successfully solved problems in other sciences such as physics, computer sciences, and biology. This project concerns the application of algebraic combinatorics, in particular its subfield the theory of symmetric functions, to solve such problems. This project is centered around the tools used, namely, the theory of symmetric functions and the associated combinatorics. Various complexity problems in representation theory concern the computation of certain structure constants and multiplicities that are expressible via the Kronecker and plethystic coefficients of the symmetric group, which can be defined using Schur functions. In statistical mechanics, the partition functions of some integrable lattice models like lozenge tilings are often Lie group characters, and their asymptotic study reveals probabilistic behavior like Gaussian unitary ensemble eigenvalue distribution near the boundary or the existence of limit shapes and surfaces. This project aims to expand these applications to study other models and distributions. Studying combinatorial and algebraic properties of Schubert polynomials, as representatives of the cohomology classes of flag varieties, can lead to combinatorial interpretations for the corresponding structure constants. Further, computational properties of their stable versions, the Stanley symmetric functions, could lead to understanding of the mysterious limit behavior of random sorting networks, corresponding to the reduced decompositions of permutations into adjacent transpositions.