Combinatorics and Complexity of Kronecker coefficients

克罗内克系数的组合学和复杂性

• 批准号: 1363193
• 负责人: Igor Pak
• 金额: $ 15万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363193&HistoricalAwards=false
• 关键词: Combinatorics; Complexity; Kronecker; coefficients

This research project concerns studies of fundamental symmetries in mathematics, and how they interact with each other, creating larger symmetries. The aim is to give a quantitative analysis of the number of times some symmetries appear as a result of such interactions. The analysis will help elucidate the nature of symmetries in general and will apply to other spaces and families of discrete objects exhibiting such symmetries. The work has important applications in probability and statistics.The proposal is concerned with the study of Kronecker coefficients of the symmetric group, which are some of the most classical objects in Algebraic Combinatorics. It aims at applications of enumerative and computational complexity nature, for estimating the coefficients and deciding their positivity. On a combinatorics side, the proposal aims to establish new unimodality results for various classes of partitions. The employed tools are largely intrinsic in these fields, involve group representation theory, Schur functions, and combinatorics of Young tableaux, with some bijective and analytic methods in partition theory added to the mix.

该研究项目涉及数学中的基本对称性以及它们如何相互作用，从而产生更大的对称性的研究。目的是对由于这种相互作用而出现的某些对称性的次数进行定量分析。该分析将有助于阐明一般对称性的本质，并将适用于表现出这种对称性的其他空间和离散物体族。 这项工作在概率和统计方面有重要的应用。该提案涉及对称群的克罗内克系数的研究，这是代数组合学中最经典的对象之一。它的目标是枚举和计算复杂性的应用，用于估计系数并确定其积极性。在组合方面，该提案旨在为各类划分建立新的单峰结果。所使用的工具很大程度上是这些领域固有的，包括群表示理论、Schur 函数和 Young tableaux 的组合数学，以及分配理论中的一些双射和解析方法。