Combinatorial Probability and Representation Theory

概率论是对事件发生或命题成立的可能性的数学研究。表示理论是通过将代数结构的元素实现为向量空间或模块上的线性映射并将其分解为最小的组成部分来研究代数结构的。概率论和表示论都适用于组合分析。例如，在概率论中，事件可以是离散状态，而在表示理论中，表示的索引集通常是一个组合对象，如分区。本建议的主要前提是发展新的组合技术来回答概率论和表示理论中的基本问题。这个项目也有大量的计算组件（贡献和使用快速增长的开源数学软件SAGEMATH）。这项研究的结果将影响超出其原始动机的领域。例如，它涉及不变量理论、复杂性理论和投票程序。这些研究是由大量的计算实验推动的。从该项目派生的算法的健壮实现将导致SAGEMATH计算机代数系统的新开源代码。这种新的SAGEMATH软件的传播不仅将推进拟议的研究计划，而且将（并且已经）使数学和计算机科学的各个领域相互促进。此外，该项目将支持多样化的研究生群体，并鼓励女学生和研究人员的参与。PI还计划组织研讨会，参与促进开放获取出版，并通过系列讲座传播结果。该项目旨在解决组合概率和表示理论中五个主题的开放性问题。五个提议的项目将与不同的合作者团队一起进行攻击，包括：(1)使用最近开发的计算任何有限马尔可夫链平稳分布的新理论来混合马尔可夫链的时间。这种方法使用了半群理论中的复杂方法。(2) k理论中关于典型稳定Grothendieck多项式的钩值表的一种新的解拥挤算法。(3)与稳定格罗滕迪克多项式相关的晶体的推广。这种方法可能会产生新的k理论插入算法。(4)分区代数及其子代数的插入算法。划分代数是一个中心化代数。在各种情况下，需要新的插入算法来匹配底层表示理论。(5)晶体碱的对称链分解可以回答体积系数的问题，在物理学、不变性理论和复杂性理论中都有应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Probability is the mathematical study of how likely an event occurs or a proposition is true. Representation theory is the study of algebraic structures by realizing their elements as linear maps on vector spaces or modules and decomposing them into their smallest constituents. Both probability and representation theory lend themselves to combinatorial analysis. For example, in probability the events could be discrete states and in representation theory the index set of a representation is often a combinatorial object such as a partition. The main premise of this proposal is the development of new combinatorial techniques for answering fundamental questions in both probability and representation theory. This project also has a substantial computational component (contributing to and using the rapidly growing open-source mathematical software SAGEMATH). The outcome of this research will impact areas beyond those of their original motivation. For example, it relates to invariant theory, complexity theory, and voting procedures. These investigations are fueled by extensive computational experimentation. Robust implementation of algorithms derived from the project will lead to new, open-source code for the SAGEMATH computer algebra system. The dissemination of this new SAGEMATH software will not only advance the proposed research program, but will (and has already) cross-fertilize various areas in mathematics and computer science. In addition, the project will support a diverse group of graduate students and encourage the participation of female students and researchers. The PI also plans to organize workshops, engage in promoting open access publishing, and disseminate results via lecture series.The project is aimed at solving open problems on five topics in combinatorial probability and representation theory. The five proposed projects will be attacked with various teams of collaborators and include: (1) Mixing times of Markov chains using the recently developed new theory for computing stationary distributions for any finite Markov chain. This approach uses sophisticated methods in semigroup theory. (2) A new uncrowding algorithm for hook-valued tableaux in relation to canonical stable Grothendieck polynomials in K-theory. (3) The generalization of a crystal associated to stable Grothendieck polynomials. This approach will likely yield new K-theoretic insertion algorithms. (4) Insertion algorithms for the partition algebra and its subalgebras. The partition algebra is a centralizer algebra. New insertion algorithms are required to match the underlying representation theory in various cases. (5) Symmetric chain decompositions of crystal bases can answer questions about plethysm coefficients and have applications to physics, invariant theory and complexity theory.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.