Combinatorial Objects and Actions in Higher Dimensional Settings

该项目由组合学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。动力学是研究系统如何变化的，是数学和应用的基础，而组合学则探索由离散结构组成的美丽的数学对象。通常，组合学和动力学提供了对数学和物理对象结构的洞察，揭示了隐藏的对称性和联系。在发现数学对象和行为方面已经取得了很大的成功，这些数学对象和行为显示出非常优雅的特性，但其中许多对象是二维的。这个提议的主题是在数学和统计物理的更高维度领域推进这些发现。这项研究还将包括指导学生和早期职业研究人员，特别是那些在数学中代表性不足的群体。这项工作包括几个涉及复杂的双射和行动的项目，建立在动态代数组合和对称函数的结果基础上。一个项目在某些类型的标签偏序集之间使用了一种新的双射，这种双射将提升和行移动的动作关联起来。另一个涉及交替符号矩阵，这是与统计物理学的方形冰模型相关的多维加泰罗尼亚类似物。这一直是一个长期存在的开放问题，以建立一个明确的双射之间的这些和某些平面分区。该项目旨在通过将这些对象解释为不同类型的白日梦，与舒伯特多项式有关的对象来解决这个问题。最后，对称群的不可约表示的网基在量子链不变量、簇代数和正几何中有应用。这项工作在更高维度的情况下寻找这样的基础，与交替符号矩阵有着令人惊讶的联系。这个奖项反映了NSF的法定使命，并被认为值得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来支持。

This project is jointly funded by the Combinatorics program and the Established Program to Stimulate Competitive Research (EPSCoR). Dynamics is the study of how systems change and is fundamental to mathematics and applications, while combinatorics explores beautiful mathematical objects composed of discrete structures. Often, combinatorics and dynamics provide insight into the structure of mathematical and physical objects, revealing hidden symmetries and connections. There has been much success in the discovery of mathematical objects and actions that display extremely elegant properties, but many of these objects are two-dimensional. The theme of this proposal is to advance such findings in higher dimensional realms of mathematics and statistical physics. This study will also include mentoring students and early career researchers, especially those of underrepresented groups in mathematics.This work is comprised of several projects involving intricate bijections and actions, building upon results in dynamical algebraic combinatorics and symmetric functions. One project uses a new bijection between certain types of labelled posets that correlates the actions of promotion and rowmotion. Another involves alternating sign matrices, which are multidimensional Catalan analogues related to the square ice model of statistical physics. It has been a long-standing open problem to construct an explicit bijection between these and certain plane partitions. This project aims to make progress on this problem by interpreting these objects as different types of pipe dreams, objects with connections to Schubert polynomials. Finally, web bases for irreducible representations of symmetric groups have applications to quantum link invariants, cluster algebras, and positive geometries. This work searches for such bases in higher dimensional cases, with surprising connections to alternating sign matrices.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.