Combinatorics and Complexity of Kronecker coefficients

这个研究项目涉及数学中基本对称性的研究，以及它们如何相互作用，创造更大的对称性。目的是定量分析由于这种相互作用而出现的一些对称性的次数。分析将有助于阐明一般的对称性的性质，并将适用于其他空间和家庭的离散对象表现出这种对称性。 这项工作在概率和统计中有重要的应用。这项建议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。它的目的是在枚举和计算复杂性的性质的应用，估计系数，并决定他们的积极性。在组合学方面，该提案旨在为各类分区建立新的单峰结果。所采用的工具在很大程度上是内在的这些领域，涉及群表示理论，舒尔函数，和组合数学的杨tableaux，与一些双射和分析方法在分区理论添加到混合。

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.