Chromatic Symmetric Functions: Solving Algebraic Conjectures Using Graph Theory

色对称函数：使用图论解决代数猜想

• 批准号: RGPIN-2022-03093
• 负责人: Crew, Logan
• 金额: $ 1.38万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749562
• 关键词: Chromatic; Symmetric; Functions; Solving; Algebraic

Many processes depend largely on the innate symmetries of natural objects, and how the movement of these objects in space affects them; for example, determining the energy states of a subatomic particle and predicting transitions between them, or distinguishing communication signals from random noise. One way to study such objects is to perturb them by rotations and analyze the resulting interactions. Because of symmetry, some rotations will produce an identical result to others. These symmetries may be encoded by symmetric functions, multivariate polynomials that are fixed under any permutation of the variables. The study of different objects can then be represented by considering symmetric functions within the context of a given algebraic structure or theory. As examples, for subatomic particles and quantum physics we consider the Fock space, and for signal identification we consider random matrix theory. The long-term goal of this research is to study symmetric functions through the theory of graphs. Graphs arise frequently in research to model discrete systems and their relationships. Graph theory is one of the most studied areas of discrete mathematics, and the main approach to this research will be to interpret symmetric functions in terms of graphs and their structure to apply a much broader range of knowledge. A starting point will be the chromatic symmetric function X_G, currently the only major connection between graphs and symmetric functions. This program will expand our knowledge helping us to understand the information X_G encodes about graphs, and to generalize X_G to a stronger form based on K-theory and other ideas from modern algebra. Additionally, this program will create graph-based constructions to represent algebraic objects such as LLT polynomials and plethysms of Schur functions, thus providing new avenues of attack for notoriously difficult problems in algebraic combinatorics such as interpreting Macdonald polynomials, and determining branching rules for representations of symmetric groups. The long-term vision of this research anticipates that these two paths will converge and build upon each other: chromatic symmetric functions are already related in special cases to LLT polynomials, which in turn are used in general symmetric function theory to study the movement of particles between quantum states in quantum mechanics. Thus, this research aims to form a bridge connecting two well-studied areas of mathematics by building from both sides. Such a bridge will greatly advance results and approaches in both fields, as techniques from one may be easily applied to the other. Students taking part in this program will learn valuable skills at the forefront of two major areas of discrete mathematics with immediate applications to positions in academia or the computer science industry.

许多过程在很大程度上取决于自然物体固有的对称性，以及这些物体在空间中的运动如何影响它们；例如，确定亚原子粒子的能量状态并预测它们之间的转变，或者区分通信信号和随机噪声。研究此类物体的一种方法是通过旋转扰动它们并分析由此产生的相互作用。由于对称性，某些旋转会产生与其他旋转相同的结果。这些对称性可以通过对称函数、在变量的任何排列下固定的多元多项式来编码。然后，可以通过在给定代数结构或理论的背景下考虑对称函数来表示对不同对象的研究。例如，对于亚原子粒子和量子物理学，我们考虑福克空间，对于信号识别，我们考虑随机矩阵理论。这项研究的长期目标是通过图论来研究对称函数。 图经常出现在对离散系统及其关系进行建模的研究中。图论是离散数学研究最多的领域之一，这项研究的主要方法是用图及其结构来解释对称函数，以应用更广泛的知识。起点是色对称函数 X_G，它是目前图和对称函数之间唯一的主要连接。该程序将扩展我们的知识，帮助我们理解 X_G 编码的图信息，并根据 K 理论和现代代数的其他思想将 X_G 推广到更强大的形式。此外，该程序将创建基于图形的构造来表示代数对象，例如 LLT 多项式和 Schur 函数的体积，从而为代数组合中众所周知的困难问题（例如解释麦克唐纳多项式和确定对称群表示的分支规则）提供新的攻击途径。这项研究的长期愿景预计这两条路径将汇聚并相互构建：色对称函数在特殊情况下已经与 LLT 多项式相关，而 LLT 多项式又被用于一般对称函数理论中，以研究量子力学中粒子在量子态之间的运动。因此，本研究旨在通过双方建设，形成一座连接两个数学研究领域的桥梁。这样一座桥梁将极大地推进这两个领域的成果和方法，因为一个领域的技术可以很容易地应用于另一个领域。参加该计划的学生将学习离散数学两个主要领域前沿的宝贵技能，并立即应用于学术界或计算机科学行业的职位。