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，到物质的状态，到矩阵和微分算子，系统的特征及其演化都是按对称性分类的。代数组合学通过对称性在众所周知的离散对象（如图形、排列和分区）中的表现来研究对称性。它的方法成功地解决了物理学、计算机科学和生物学等其他科学中的问题。该项目涉及应用代数组合学，特别是其子领域对称函数理论来解决此类问题。该项目以所使用的工具为中心，即对称函数理论和相关的组合数学。表示论中的各种复杂性问题涉及某些结构常数和多重性的计算，这些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，可以使用 Schur 函数来定义。在统计力学中，一些可积晶格模型（如菱形镶嵌）的配分函数通常具有李群特征，其渐近研究揭示了概率行为，如边界附近的高斯酉系综特征值分布或极限形状和曲面的存在。该项目旨在扩展这些应用程序以研究其他模型和分布。研究舒伯特多项式的组合和代数性质，作为标志簇上同调类的代表，可以得出相应结构常数的组合解释。此外，其稳定版本（斯坦利对称函数）的计算特性可以帮助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减少分解。