{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.08033",
  "author_total": 1,
  "authors": [
    {
      "name": "Colin Defant"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dghs2244z0h3b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.08033",
  "fulltext_url": null,
  "id": "dghs2244z0h3b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.08033",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7160833468",
  "pdf_url": "https://arxiv.org/pdf/2605.08033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dghs2244z0h3b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2605.08033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.08033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Weak Order on the MacNeille Completion of Bruhat Order",
  "title_text": "Weak Order on the MacNeille Completion of Bruhat Order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/dghs2244z0h3b",
  "venue": "arXiv",
  "versions": []
}