Bruhat序的MacNeille完备性的弱序
Weak Order on the MacNeille Completion of Bruhat Order
AI总结：
本文引入了0-Hecke monoid作用于MacNeille完备性，定义弱序和下降集统计，并证明了某些Knutson-Miller子词复形的顶点可分解性，以及Cohen-Macaulay ASM变种的猜想。
摘要：
设Mac(W)为Coxeter群W的Bruhat序的MacNeille完备性。我们引入了W类型的0-Hecke monoid作用于Mac(W)，从而定义了弱序和下降集统计。当W为类型A时，我们恢复了Hamaker和Reiner的构造，最初是用单调三角形和交替符号矩阵表述的。利用这一作用，我们证明了某些Knutson-Miller子词复形的顶点可分解性。通过专精于类型A，我们证明了Escobar、Klein和Weigandt关于Cohen-Macaulay ASM变种的猜想。在此过程中，我们还展示了Hamaker和Reiner关于ASM弱序区间poset拓扑猜想的反例。最后，当W为有限不可约时，我们使用0-Hecke作用在Mac(W)上引入了一个非可逆的动力系统，称为MacNeille pop-stack算子，并证明了达到底部状态所需最大迭代次数为h-1，其中h是W的Coxeter数。本文旨在作为使用大型语言模型自动化数学研究工作流的案例研究。Escobar-Klein-Weigandt猜想的证明和Hamaker-Reiner猜想的反驳由ChatGPT 5.4 Pro自主完成。其他论文部分主要由作者完成，但ChatGPT加速了进程。我们提供了详细的互动记录，并推测了模型成功的原因。
英文摘要：
Let $\mathrm{Mac}(W)$ be the MacNeille completion of the Bruhat order of a Coxeter group $W$. We introduce an action of the $0$-Hecke monoid of type $W$ on $\mathrm{Mac}(W)$, which allows us to define a weak order and a descent set statistic on $\mathrm{Mac}(W)$. When $W$ is of type $A$, we recover constructions of Hamaker and Reiner, which were originally formulated in terms of monotone triangles and alternating sign matrices. Using this action, we prove that certain unions of Knutson--Miller subword complexes are vertex-decomposable. By specializing to type $A$, we prove a conjecture of Escobar, Klein, and Weigandt regarding Cohen--Macaulay ASM varieties. Along the way, we also exhibit a counterexample to a conjecture of Hamaker and Reiner regarding the poset topology of intervals in the ASM weak order. When $W$ is finite, our action also gives rise to a Boolean cell complex with one facet for each element of $\mathrm{Mac}(W)$. We prove that this is an abstract simplicial complex and that every linear extension of the weak order on $\mathrm{Mac}(W)$ is a shelling order for its facets. Finally, when $W$ is finite and irreducible, we use our $0$-Hecke action to introduce a noninvertible dynamical system on $\mathrm{Mac}(W)$ that we call the \emph{MacNeille pop-stack operator}, and we prove that the maximum number of iterations of this operator needed to reach the bottom state is $h-1$, where $h$ is the Coxeter number of $W$. This article is meant to serve as a case study in using large language models to automate the workflow of mathematical research. The proof of the conjecture of Escobar--Klein--Weigandt and the disproof of the conjecture of Hamaker--Reiner were obtained autonomously by ChatGPT 5.4 Pro. Other aspects of the paper were obtained mostly by the author, but ChatGPT expedited the process. We provide a detailed account of this interaction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
