Combinatorics and Asymptotics of Structure Constants from Representation Theory and Algebra

对称性和模式捕捉了我们复杂世界的本质，并给出了用代数和离散方法研究它所需的抽象。随着物体的变化和相互作用，固有的对称性也会发生变化。对称如何组合、限制、投射或转换成其他对称？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--每种类型的多少个组件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一般表明不会存在“封闭形式”的答案，但这个项目的目标是找到这些数字并查看典型结构是什么样子的。这些问题以许多伪装的形式出现，涉及组合学、代数、表示论、概率统计力学和计算复杂性理论。更准确地说，PI的目的是解决代数组合学中涉及“结构常数”和Young图的问题。结构常数一般被定义为张量积或合成分解中不可约对称或一般线性群表示的重数，或者更一般地，各种对称函数在某些基上分解时的非负整系数。PI的目的是确定这些常量的行为--渐近性、正性、相互关系、组合解释。在旗舰问题和最终目标中有：Kronecker系数和体积系数的组合解释，Foulkes关于体积系数相对顺序的猜想，Littlewood-Richardson和Kronecker系数的渐近性，参数在不同生长区域中的斜(半)标准Young图面的渐近数目，“斜”(一般的，非梯形)区域的菱形平铺的极限行为，几何复杂性理论中导致多项式之间的障碍的重数的不等式，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。这些方法的范围从PI工作中现有方法的扩展，到与统计力学方法的进一步结合，如变分原理、大偏差枚举法、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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.