Weak Order on the MacNeille Completion of Bruhat Order

Let $\mathrm{M}\mathrm{a}\mathrm{c}(W)$ 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 $\mathrm{M}\mathrm{a}\mathrm{c}(W)$, which allows us to define a weak order and a descent set statistic on $\mathrm{M}\mathrm{a}\mathrm{c}(W)$. 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. Finally, when $W$ is finite and irreducible, we use our $0$-Hecke action to introduce a noninvertible dynamical system on $\mathrm{M}\mathrm{a}\mathrm{c}(W)$ that we call the 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, and we speculate on what allowed the model to be successful.