Combinatorics and Asymptotics of Structure Constants from Representation Theory and Algebra

来自表示论和代数的结构常数的组合学和渐近学

• 批准号: 1800423
• 负责人: Greta Panova
• 金额: $ 15万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800423&HistoricalAwards=false
• 关键词: Combinatorics; Asymptotics; Structure; Constants; Representation

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.

对称性和模式捕捉了复杂世界的本质，并提供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。对称性如何组合、限制、投射或转化为其他对称性？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——另一个更大的对称结构中包含每种类型的多少个组件。虽然问题的计算复杂性通常暗示不存在“封闭形式”答案，但该项目的目标是近似找到这些数字并了解典型结构的外观。这些问题以多种形式出现，并且存在于组合学、代数、表示论、概率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”和杨氏造型的代数组合问题。结构常数通常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重数，或者更一般地定义为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。旗舰问题和最终目标包括：克罗内克和体积系数的组合解释、福克斯关于体积系数相对顺序的猜想、Littlewood-Richardson 和克罗内克系数的渐近性、偏斜（半）标准杨图的渐近数参数的各种生长方式，“倾斜”菱形平铺的限制行为（一般，非梯形）域、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、色对称函数的 q 类似物的 e 展开式的正性以及 LLT 多项式的 Schur 正性。这些方法包括对 PI 工作中现有方法的扩展，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，并被认为是值得的。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来提供支持。