Algebraic, Combinatorial, and Analytic Applications of Symmetric Functions

对称函数的代数、组合和解析应用

• 批准号: 1500834
• 负责人: Greta Panova
• 金额: $ 13万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500834&HistoricalAwards=false
• 关键词: Algebraic; Combinatorial; Analytic; Applications; Symmetric

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.

从晶体的晶格结构到物质状态，再到矩阵和差分运算符，系统的特征及其进化都通过对称性进行了分类。代数组合学通过其表现形式在图形，排列和分区等众所周知的离散对象中进行的表现。它的方法已成功地解决了其他科学，例如物理，计算机科学和生物学。该项目涉及代数组合学的应用，尤其是其子领域的对称函数理论来解决此类问题。该项目围绕使用的工具，即对称函数的理论和相关的组合学理论。表示理论中的各种复杂性问题涉及通过对称组的Kronecker和多个系数表达的某些结构常数和多重性的计算，可以使用Schur函数来定义。在统计力学中，某些可集成的晶格模型（如lozenge砖块）的分区功能通常是谎言群体，它们的渐近研究揭示了概率行为，例如高斯单位统一的集合特征值分布在边界附近或极限形状和表面的存在。该项目旨在将这些应用程序扩展到研究其他模型和分布。研究舒伯特多项式的组合和代数特性，作为国旗品种的同种学类别的代表，可以导致对相应结构常数的组合解释。此外，其稳定版本的计算特性，史丹利对称函数可能会导致对随机排序网络的神秘极限行为的理解，这对应于置换分解减少到相邻的换位中。