Kostka, Littlewood-Richardson, Kronecker, and plethysm coefficients are fundamental quantities in algebraic combinatorics, yet many natural questions about them stay unanswered for more than 80 years. Kronecker and plethysm coefficients lack ``nice formulas'', a notion that can be formalized using computational complexity theory. Beyond formulas and combinatorial interpretations, we can attempt to understand their asymptotic behavior in various regimes, and inequalities they could satisfy. Understanding these quantities has applications beyond combinatorics. On the one hand, the asymptotics of structure constants is closely related to understanding the [limit] behavior of vertex and tiling models in statistical mechanics. More recently, these structure constants have been involved in establishing computational complexity lower bounds and separation of complexity classes like VP vs VNP, the algebraic analogs of P vs NP in arithmetic complexity theory. Here we discuss the outstanding problems related to asymptotics, positivity, and complexity of structure constants focusing mostly on the Kronecker coefficients of the symmetric group and, less so, on the plethysm coefficients. This expository paper is based on the talk presented at the Open Problems in Algebraic Combinatorics coneference in May 2022.

翻译：Kostka系数、Littlewood-Richardson系数、Kronecker系数以及plethysm系数是代数组合学中的基本量，然而关于它们的许多自然问题在80多年来仍未得到解答。Kronecker系数和plethysm系数缺乏"优美公式"——这一概念可通过计算复杂性理论予以形式化。除了公式和组合解释之外，我们还可以尝试理解它们在不同体系下的渐近行为，以及它们可能满足的不等式。理解这些量具有超越组合学的应用价值。一方面，结构常数的渐近行为与统计力学中顶点模型和拼贴模型的极限行为密切相关。近年来，这些结构常数还被用于建立计算复杂性下界以及区分复杂性类（如VP与VNP，后者是算术复杂性理论中P与NP的代数类比）。本文主要围绕对称群的Kronecker系数（并简要涉及plethysm系数），讨论与结构常数的渐近性、正定性和复杂性相关的若干未解决问题。这篇综述性文章基于2022年5月在"代数组合学开放问题"会议上所作的报告。