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arxiv: 2605.08033 · v1 · submitted 2026-05-08 · 🧮 math.CO

Recognition: no theorem link

Weak Order on the MacNeille Completion of Bruhat Order

Colin Defant

Pith reviewed 2026-05-11 02:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords MacNeille completionBruhat order0-Hecke monoidweak orderCoxeter numberalternating sign matricespop-stack operatorsubword complexes
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The pith

The 0-Hecke monoid acts on the MacNeille completion of the Bruhat order to define a weak order whose pop-stack operator reaches the bottom in at most h-1 steps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an action of the 0-Hecke monoid of type W on the MacNeille completion Mac(W) of the Bruhat order on a Coxeter group W. This action is compatible with the lattice structure and thereby defines a weak order and descent statistics on the completion, recovering the known type-A constructions that use monotone triangles and alternating sign matrices. The same action proves that unions of Knutson-Miller subword complexes are vertex-decomposable and, when specialized to type A, confirms that certain associated varieties are Cohen-Macaulay. For any finite irreducible W the induced MacNeille pop-stack operator requires at most h-1 iterations to reach the minimal element, where h is the Coxeter number. Along the way a counterexample is given to a conjecture on the topology of intervals in the alternating-sign-matrix weak order.

Core claim

The 0-Hecke monoid of type W acts on the MacNeille completion Mac(W) of the Bruhat order on W. This action respects the lattice operations and therefore induces a well-defined weak order together with descent sets on Mac(W). When W is finite and irreducible the associated MacNeille pop-stack operator reaches the bottom element after at most h-1 steps, h being the Coxeter number of W. Specializing to type A recovers the Hamaker-Reiner structures on alternating sign matrices, proves the Escobar-Klein-Weigandt conjecture on Cohen-Macaulay ASM varieties, and supplies a counterexample to the Hamaker-Reiner conjecture on the poset topology of ASM intervals.

What carries the argument

The action of the 0-Hecke monoid of type W on Mac(W), which extends the usual weak-order action while remaining compatible with the meet and join operations of the MacNeille completion lattice.

If this is right

  • Unions of Knutson-Miller subword complexes are vertex-decomposable.
  • The Escobar-Klein-Weigandt conjecture holds: the relevant ASM varieties are Cohen-Macaulay.
  • A counterexample exists to the Hamaker-Reiner conjecture on the topology of intervals in the ASM weak order.
  • The maximum number of MacNeille pop-stack iterations is bounded by h-1 for every finite irreducible Coxeter group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a uniform combinatorial model for weak-order phenomena that applies to every Coxeter type rather than only type A.
  • The bound involving the Coxeter number suggests that the height of the new dynamics is governed by the same invariant that controls the length of reduced words and the degrees of the coinvariant algebra.
  • The vertex-decomposability result may be used to compute the homotopy type of the relevant subword complexes by induction on the number of complexes in the union.

Load-bearing premise

The 0-Hecke monoid action on the MacNeille completion must be well-defined and must preserve the lattice operations so that the induced weak order relations remain consistent.

What would settle it

An explicit element of Mac(W) for some finite irreducible Coxeter group W that requires more than h-1 iterations of the MacNeille pop-stack operator to reach the bottom element.

Figures

Figures reproduced from arXiv: 2605.08033 by Colin Defant.

Figure 1
Figure 1. Figure 1: An interval [A′ , A]L in the ASM weak order on ASM6 such that the order complex ∆L(A′ , A) is not contractible. Below each alternating sign matrix B is the set min(Ψ(B)) ⊆ S6. which have been studied extensively ever since [BC17,EM18,PS15,STW25]. For F ∈ Mac(W), we consider the simplicial complex ∆(Q, F) = [ w∈min(F) ∆(Q, w), where min(F) is the set of minimal elements of F (in Bruhat order) [PITH_FULL_IM… view at source ↗
read the original 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. 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript defines Mac(W) as the MacNeille completion of the Bruhat order on a Coxeter group W. It introduces an action of the 0-Hecke monoid of type W on Mac(W) to extend the weak order and descent set statistic. For type A this recovers prior constructions on alternating sign matrices and monotone triangles. The action is used to prove that certain unions of Knutson-Miller subword complexes are vertex-decomposable, to establish a conjecture of Escobar-Klein-Weigandt on the Cohen-Macaulay property of ASM varieties, and to exhibit a counterexample to a conjecture of Hamaker-Reiner on the topology of intervals in the ASM weak order. For finite irreducible W the paper defines the MacNeille pop-stack operator via the action and proves that its iteration count to the bottom element is at most the Coxeter number h-1. The work is presented as a case study in LLM-assisted mathematical research, with the conjecture proof and counterexample generated autonomously by ChatGPT 5.4 Pro.

Significance. If the 0-Hecke action is shown to be a well-defined monoid homomorphism on the full lattice Mac(W) and the key derivations are independently verified, the results would supply new combinatorial machinery for studying completions of Bruhat order, resolve an open question on ASM varieties, and introduce a dynamical system whose period is controlled by the Coxeter number. The counterexample and vertex-decomposability statements would also contribute concrete advances in poset topology and subword complex theory.

major comments (2)
  1. [section defining the 0-Hecke action] The section introducing the 0-Hecke monoid action on Mac(W): the manuscript must explicitly verify that the generators satisfy T_s^2 = T_s and the braid relations when applied to meets and joins that are added by the MacNeille completion and are absent from the original Bruhat poset. Without this verification the descent sets are not unambiguously defined and the subsequent dynamical system is not guaranteed to be well-defined.
  2. [final section on the pop-stack operator] The proof that the MacNeille pop-stack operator reaches the bottom in at most h-1 steps (final section): this bound is load-bearing for the main dynamical claim and rests on the lattice compatibility of the action; an explicit check for a small irreducible group (e.g., type A_2 or B_2) together with the general argument would be required to confirm that no extra relations arise on the added elements.
minor comments (2)
  1. [abstract and type-A specialization paragraph] The abstract states that the type-A case recovers Hamaker-Reiner constructions; a short explicit dictionary between the new descent sets and the classical ones on ASMs would improve readability.
  2. [final discussion section] The account of LLM interaction is presented as a case study; adding the precise prompts that produced the conjecture proof and counterexample would allow readers to assess reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. The comments identify important points that require clarification and additional verification to ensure the 0-Hecke action and the associated dynamical system are rigorously well-defined on the MacNeille completion. We address each major comment point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [section defining the 0-Hecke action] The section introducing the 0-Hecke monoid action on Mac(W): the manuscript must explicitly verify that the generators satisfy T_s^2 = T_s and the braid relations when applied to meets and joins that are added by the MacNeille completion and are absent from the original Bruhat poset. Without this verification the descent sets are not unambiguously defined and the subsequent dynamical system is not guaranteed to be well-defined.

    Authors: We agree that an explicit verification of the monoid relations on the added lattice elements is necessary for a complete proof. The manuscript defines the action on the Bruhat poset W and extends it to Mac(W) by requiring that the action preserve meets and joins, but we did not include a separate lemma confirming that T_s^2 = T_s and the braid relations continue to hold after this extension. In the revised version we will insert a short subsection that proves these relations hold on the full lattice by using the universal property of the MacNeille completion together with the fact that the original action on W is order-preserving and satisfies the relations. This will unambiguously define the descent sets and the dynamical system. revision: yes

  2. Referee: [final section on the pop-stack operator] The proof that the MacNeille pop-stack operator reaches the bottom in at most h-1 steps (final section): this bound is load-bearing for the main dynamical claim and rests on the lattice compatibility of the action; an explicit check for a small irreducible group (e.g., type A_2 or B_2) together with the general argument would be required to confirm that no extra relations arise on the added elements.

    Authors: We accept that an explicit verification for small-rank cases would strengthen the argument and rule out unforeseen relations on the added elements. The general proof in the manuscript relies on the lattice compatibility established earlier, but we will augment the final section with direct computations for the irreducible groups of types A_2 and B_2. These calculations will confirm that the pop-stack operator reaches the bottom element in at most h-1 steps on the full MacNeille lattice and that no additional relations appear. The explicit checks will be presented as supporting evidence for the general bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions are self-contained via explicit definitions and standard prior results

full rationale

The paper defines the 0-Hecke monoid action on Mac(W) explicitly to ensure it is a monoid homomorphism compatible with meets and joins in the MacNeille completion, then uses this to define weak order, descent sets, and the pop-stack operator. These steps rely on the standard definitions of MacNeille completion and 0-Hecke monoids from external literature, with the compatibility verified directly rather than assumed or fitted. The bound of h-1 iterations follows from the Coxeter number and the action's properties without reducing to self-citation chains or renaming inputs as outputs. The proofs of the cited conjectures are derived within this framework as independent consequences, with no load-bearing step collapsing to a prior result by the same authors or by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard definitions and properties of Coxeter groups, Bruhat order, MacNeille completions of posets, and 0-Hecke monoids; no free parameters are fitted to data.

axioms (2)
  • standard math The MacNeille completion of any poset is a complete lattice containing the original poset as a subposet.
    Invoked to define Mac(W) from the Bruhat order.
  • domain assumption The 0-Hecke monoid of type W acts on the Bruhat order in a manner compatible with the lattice operations of the MacNeille completion.
    Required for the action to extend and define the weak order on Mac(W).
invented entities (1)
  • MacNeille pop-stack operator no independent evidence
    purpose: Noninvertible dynamical system on Mac(W) obtained by iterating the 0-Hecke action until the bottom element is reached.
    Newly defined in the paper for finite irreducible W.

pith-pipeline@v0.9.0 · 5643 in / 1700 out tokens · 46972 ms · 2026-05-11T02:14:58.933941+00:00 · methodology

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Reference graph

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