Tensor and wreath products of symmetric groups

Kronecker系数和Plethysm系数描述了对称函数的乘积分解成它们的简单分量。它们在数学中无处不在，就像对称的概念本身一样。Kronecker系数被描述为“也许是代数组合学中最具挑战性、最深刻和最神秘的对象”。它们在对称函数理论以及一般线性和对称群的表示理论中扮演着重要的角色。尽管研究了80年，但“令人沮丧的是，人们对这些系数知之甚少”。我希望通过首先考虑它们看起来是“一般的”或“直到有限不稳定的”，来理解这些克朗尼克系数和多体系数的整个蓝图。我最近开创了一种新的方法，在配分代数的背景下理解这些系数的(稳定)蓝图，从而完整地描述了Kronecker系数的稳定蓝图的一半，并归纳地描述了多体系数的稳定蓝图。在这一成功的基础上，这一建议试图完全理解稳定的Kronecker系数和体积系数，并解决关于非稳定Kronecker系数和体积系数的正性的旧的和新的猜想。

The Kronecker and plethysm coefficients describe the decompositions of products of symmetric functions into their simple constituents. They are as ubiquitous across mathematics as the notion of symmetry itself. The Kronecker coefficients have been described as "perhaps the most challenging, deep and mysterious objects in algebraic combinatorics''. They play an important role in the theory of symmetric functions and in the representation theory of general linear and symmetric groups. Despite 80 years of study, "frustratingly little is known'' about these coefficients. I hope to understand the whole blueprint for these Kronecker and plethysm coefficients by first considering what they looks like "generically" or "up to a finite instability". I have recently pioneered a new approach to understanding the (stable) blueprints of these coefficients in the context of the partition algebra and hence completely described one half of the stable blueprint for Kronecker coefficients and inductively described the stable blueprint for plethysm coefficients. Building on this success, this proposal seeks to completely understand the stable Kronecker and plethysm coefficients and to solve old and new conjectures concerning the positivity properties of non-stable Kronecker and plethysm coefficients.