Equivariant Combinatorics

“等变组合数学”一词是在2017年夏季蒙特利尔的一次会议上提出的。等变映射是与对称性交织的映射。在研究离散对象的组合学中，它也可以意味着映射交织了组合对象或与之相关的表示的某些统计。广义上，它描述了代数组合学，代数几何，代数拓扑，表示论和理论物理之间的深刻相互作用，因为这些学科都致力于研究自然和数学中的对称性。因此，这个短语也包含了通过表示论与概率和半群理论的联系。本项目的主要前提是探索和进一步发展这些新的关系。该项目也有一个重要的计算部分（贡献和使用快速增长的开源数学软件SageMath）。该项目旨在解决五个开放的等变组合问题。他们将与各种团队的合作者进行攻击，包括：（1）研究晶体基础，解释一般线性群的特征理论中的体积。这在物理学、不变量理论和复杂性理论中都有应用。(2)LLT多项式是组合数学和几何学中重要的对称函数，本文对LLT多项式进行了组合分析。这涉及到对潜在组合对象的新统计。(3)证明Shimozono-Weyman猜想。这依赖于一个与kattero兼容的符号反转对合。(4)与李超代数相关的晶体。这需要发展起源于弦理论的李超代数的不可约晶体理论。(5)Tsetlin图书馆的统一理论。这是一个新的方法，以理论的马尔可夫链相关的半群。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

The term "Equivariant Combinatorics" was coined at a summer 2017 conference in Montreal. An equivariant map is a map that intertwines with symmetries. In combinatorics, which is the study of discrete objects, it can also mean that the map intertwines certain statistics on combinatorial objects or the representations associated to them. Broadly, it describes profound interactions between algebraic combinatorics, algebraic geometry, algebraic topology, representation theory, and theoretical physics because these subjects are wedded to the study of symmetries in nature and mathematics. The phrase therefore also encompasses ties to probability and semigroup theory through representation theory. The main premise of this project is the exploration and further development of these new relationships. This project also has a substantial computational component (contributing to and using the rapidly growing open source mathematical software SageMath).The project is aimed at solving five open equivariant combinatorics problems. They will be attacked with various teams of collaborators and include: (1) The study of a crystal basis explaining the plethysm in the character theory of the general linear group. This has applications to physics, invariant theory and complexity theory. (2) A combinatorial analysis of LLT polynomials, which are important symmetric functions in combinatorics and geometry. This involves new statistics on the underlying combinatorial objects. (3) A proof of the Shimozono-Weyman conjecture. This relies on a sign-reversing involution compatible with katabolism. (4) Crystals associated to Lie superalgebras. This requires developing the theory of irreducible crystals for Lie superalgebras that originated in string theory. (5) A unified theory of Tsetlin libraries. This is a new approach to the theory of Markov chains associated to semigroups.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.