Combinatorics and Asymptotics of Structure Constants from Representation Theory and Algebra

对称和模式抓住了我们复杂世界的本质，并为用代数和离散方法研究它提供了必要的抽象。随着物体的变化和相互作用，固有的对称性也在变化。对称如何组合、限制、投射或转化为其他对称？我们如何分解一个由不可约成分组成的复杂系统？一般来说，我们想要定量地描述这种相互作用——每种类型的成分有多少包含在另一个更大的对称结构中。虽然这个问题的计算复杂性通常暗示不存在“封闭形式”的答案，但这个项目的目标是找到这些近似的数字，并看看典型的结构是什么样子的。这些问题以多种形式出现，并且处于组合学、代数、表示理论、概率论和统计力学以及计算复杂性理论的交叉点。更准确地说，PI旨在解决涉及“结构常数”和杨氏表的代数组合问题。结构常数通常被定义为张量积或组合分解中不可约对称或一般线性群表示的复数，或者更一般地，在某些基中各种对称函数分解中的非负积分系数。PI旨在确定这些常数的行为——渐近性、正性、相互关系、组合解释。主要问题和最终目标包括：Kronecker和plethysm系数的组合解释，Foulkes关于plethysm系数的相对顺序的猜想，Littlewood-Richardson和Kronecker系数的渐近性，参数在各种生长状态下的偏（半）标准Young表的渐近数，“偏”（一般，非梯形）域的菱形平顶的极限行为，几何复杂性理论中的多重性不等式导致多项式之间的区分障碍，颜色对称函数的q-类似物的e-展开的正性和LLT多项式的Schur正性。这些方法包括对PI现有工作方法的扩展，以及与统计力学方法的进一步结合，如变分原理、大偏差枚举、代数几何解释等。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

Symmetries and patterns capture the essence of our complex world and give the abstraction necessary to study it with algebraic and discrete methods. As objects change and interact, so do the inherent symmetries. How do symmetries combine, restrict, project or transform into other symmetries? How do we decompose a complex system of irreducible components? In general we want to describe this interaction quantitatively -- how many components of each type are contained in another bigger symmetry structure. While the computational complexity of the problem in general hints that no "closed-form" answer would exist, it is the goal of this project to find these numbers approximately and see how a typical structure looks like. These problems appear in many disguises, and lie at the intersection of combinatorics, algebra, representation theory, probability and statistical mechanics, and computational complexity theory. More precisely, the PI aims to solve problems in algebraic combinatorics involving "structure constants" and Young tableaux. Structure constants are generally defined as the multiplicities of irreducible symmetric or general linear group representations in the decomposition of tensor products or compositions, or, more generally, the nonnegtive integral coefficients in the decomposition of various symmetric functions in certain bases. The PI aims to determine the behavior of such constants -- asymptotics, positivity, relation to each other, combinatorial interpretation. Among the flagship problems and ultimate goals are: combinatorial interpretation for the Kronecker and plethysm coefficients, Foulkes' conjecture on the relative order of plethysm coefficients, asymptotics of Littlewood-Richardson and Kronecker coefficients, the asymptotic number of skew (semi)Standard Young tableaux in various growth regimes for the parameters, limit behavior of lozenge tilings of "skew" (general, nontrapezoidal) domains, inequalities of multiplicities in Geometric complexity Theory leading to obstructions distinguishing between polynomials, positivity of e-expansions for q-analogues of chromatic symmetric functions and Schur positivity for LLT polynomials. The methods range from extension of existing approaches in the PI's work, to further combination with methods from statistical mechanics like the variational principle, enumeration via large deviations, algebro-geometric interpretations, etc.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.