Equivariant Combinatorics

“等变组合学”一词是在2017年夏季蒙特利尔的一次会议上创造的。等变映射是一种与对称交织的映射。在研究离散对象的组合学中，它也可以意味着映射将组合对象或与其相关的表示上的某些统计数据交织在一起。从广义上讲，它描述了代数组合学、代数几何、代数拓扑、表示理论和理论物理之间深刻的相互作用，因为这些学科与自然界和数学中的对称性研究密切相关。因此，这个短语也包含了通过表示理论与概率论和半群理论的联系。这个项目的主要前提是探索和进一步发展这些新的关系。这个项目也有一个重要的计算组件（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等变组合问题。他们将受到不同团队的合作者的攻击，包括：(1)晶体基础的研究，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) ltt多项式的组合分析，它是组合学和几何中重要的对称函数。这涉及到底层组合对象的新统计信息。(3) Shimozono-Weyman猜想的证明。这依赖于与分解代谢相容的符号反转内联。(4)李超代数晶体。这就需要为起源于弦理论的李超代数发展不可约晶体理论。(5)蔡德林图书馆的统一理论。这是研究半群马尔可夫链理论的一种新方法。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。

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.