Combinatorial Objects and Actions in Higher Dimensional Settings

该项目由组合学计划和已建立的激励竞争研究计划(EPSCoR)共同资助。动力学是研究系统如何变化的学科，是数学和应用的基础，而组合学则研究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的结构的洞察，揭示了隐藏的对称性和联系。在发现表现出极其优雅性质的数学对象和动作方面取得了很大的成功，但这些对象中的许多都是二维的。这项提议的主题是在更高维度的数学和统计物理领域推进这些发现。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。这项工作包括几个项目，涉及复杂的双射和行动，建立在动态代数组合学和对称函数的结果的基础上。一个项目在某些类型的标记偏序集之间使用了一种新的双射，这种偏序集将提升和划船动作联系在一起。另一个涉及交替符号矩阵，这是与统计物理的正方形冰模型相关的多维加泰罗尼亚类似物。在这些平面划分和某些平面划分之间构造显式双射一直是一个悬而未决的问题。这个项目旨在通过将这些物体解释为不同类型的白日梦，与舒伯特多项式有关的物体来在这个问题上取得进展。最后，对称群的不可约表示的网基可应用于量子链接不变量、簇代数和正几何。这项工作在更高维度的案例中寻找这样的基础，与交替符号矩阵有着惊人的联系。这个奖项反映了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。

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.