Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some beautiful formulas and combinatorial interpretations. The flagship hook-length formula counts the number of Standard Young Tableaux, which also gives the dimension of the irreducible Specht modules of the Symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, that could formally explain the beauty we see and the difficulties we encounter in finding further formulas and ``combinatorial interpretations''. A 85-year-old such problem asks for a positive combinatorial formula for the Kronecker coefficients of the Symmetric group, another one pertains to the plethysm coefficients of the General Linear group. In the opposite direction, the study of Kronecker and plethysm coefficients leads to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail, possibly asymptotically.

翻译：代数组合学起源于代数学与表示论，通过组合方法研究其中的离散对象与整数型量，这些方法后来发展出独立而自洽的理论体系，并为我们带来了优美的公式与组合解释。旗舰性的钩长公式（hook-length formula）用于计数标准杨表（Standard Young Tableaux）的数量，同时给出了对称群不可约Specht模的维数。优雅的李特尔伍德-理查德森法则（Littlewood-Richardson rule）则描述了GL-模张量积中不可约GL-模的重数。这些公式与法则不仅激发了代数学与组合学领域广泛的研究与发展，还逐渐适用于可积概率论、统计力学以及计算复杂性理论。本文将探讨超出这类优美乘积公式与组合解释所能触及的范围，进入计算复杂性理论的领域——该理论能够正式解释我们所见的优美性以及进一步寻找公式与“组合解释”时遇到的困难。一个已有85年历史的问题要求为对称群的Kronecker系数给出正的组合公式；另一个问题则涉及一般线性群的plethysm系数。反过来说，对Kronecker系数与plethysm系数的研究导致了对几何复杂性理论（GCT）中试图解决代数P与NP千禧年问题（即VP与VNP问题）的乐观方法的证伪。为了使GCT得以奏效并建立计算复杂性的下界，我们需要更深入地理解表示论中的重数，或许需要在渐近意义下进行研究。