Combinatorics and Complexity of Kronecker coefficients

这项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。其目的是对某些对称性因这种相互作用而出现的次数进行定量分析。这一分析将有助于阐明对称性的一般性质，并将适用于其他空间和表现出这种对称性的离散物体族。这项工作在概率和统计学中有着重要的应用。这项工作涉及到对称群的Kronecker系数的研究，对称群是代数组合学中最经典的对象之一。它针对具有枚举性和计算复杂性的应用，用于估计系数并确定其正性。在组合学方面，该提案旨在为各种类型的划分建立新的单峰性结果。所使用的工具在很大程度上是这些领域的固有工具，涉及群表示论、舒尔函数和Young Tableaux的组合学，此外还加入了分拆理论中的一些双射和分析方法。

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.