arxiv: 2604.26502 · v1 · submitted 2026-04-29 · 🧮 math.CO

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MacNeille completions of parabolic quotients

Yibo Gao , Hanlin Xu

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keywords alternating sign matricesBruhat orderDedekind-MacNeille completionparabolic quotientssymmetric groupASM varietieslattice operationsposet completion

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The pith

The Dedekind-MacNeille completion of the Bruhat order on any parabolic quotient of the symmetric group is a subposet of alternating sign matrices whose lattice operations correspond to unions and intersections of ASM varieties.

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

This paper gives an explicit description of the Dedekind-MacNeille completion of the Bruhat order for every parabolic quotient of the symmetric group. These completions appear inside the alternating sign matrices but carry their own meet and join operations rather than the standard ones on ASMs. The authors show that these operations align with taking unions and intersections of the associated ASM varieties. A sympathetic reader would care because the result extends the known completion for the full symmetric group to a uniform combinatorial and geometric picture across all parabolic quotients.

Core claim

Alternating sign matrices arise as the Dedekind-MacNeille completion of the Bruhat order on the symmetric group. We explicitly describe the Dedekind-MacNeille completion of the Bruhat order on any parabolic quotients of the symmetric group. It is naturally a subposet of the alternating sign matrices, with different lattice operations. Moreover, we demonstrate the relations between the meet and join operations in this lattice with taking unions and intersections of the corresponding ASM varieties, respectively. Special cases receive further discussion.

What carries the argument

The Dedekind-MacNeille completion of the Bruhat order on parabolic quotients, realized as a subposet of alternating sign matrices with meet and join given by unions and intersections of ASM varieties.

Load-bearing premise

The explicitly described subposet inside alternating sign matrices is exactly the smallest lattice containing each parabolic quotient under the Bruhat order, and its operations match the unions and intersections of the corresponding ASM varieties.

What would settle it

A parabolic quotient for which the described subposet is not closed under the required meets and joins or for which those operations fail to correspond to the unions and intersections of the associated ASM varieties.

Figures

Figures reproduced from arXiv: 2604.26502 by Hanlin Xu, Yibo Gao.

**Figure 1.** Figure 1: The MacNeille completion ASMI (n) of S I n for n = 4 and I = {2} lattice if any arbitrary subset has a join and a meet. In case of finite posets, these two notions conincide. An element x ∈ P is called join-irreducible (resp. meet-irreducible) if it cannot be written as s ∨ t for s, t < x (resp. s ∧ t for s, t > x). The following is Proposition 3.3.1 in [Sta97]. Proposition 2.1. Let P be a finite poset wit… view at source ↗

**Figure 2.** Figure 2: Bijection between alternating sign matrices (left) and monotone triangles (right) a, b, c ∈ N, a + b − n ≤ c ≤ a, b, defined as P[a, b, c](i) =    i, i ≤ c or i > a + b − c, b − c + i, c < i ≤ a, −a + c + i, a < i ≤ a + b − c. Similarly, we also define the following permutations, for (a, b, c) in the same range: Q[a, b, c](i) =    n + 1 − i, i ≤ a − c or i > n − b + c, a + b − c + 1 − i, a − c < … view at source ↗

**Figure 3.** Figure 3: The join-irreducible elements P[a, b, c] (left) and meetirreducible elements Q[a, b, c] (right) following technical results can be found in [LS96]. Proposition 2.6. Let a, b, c ∈ N with a + b − n ≤ c ≤ a, b, then we have (1) rP[a,b,c](a, b) = rQ[a,b,c](a, b) = c. (2) For any M ∈ ASM(n) such that rM(a, b) = c, P[a, b, c] ≤ M ≤ Q[a, b, c]. (3) P[a, b, c] and Q[a, b, c] are join-irreducible and meet-irreduci… view at source ↗

**Figure 4.** Figure 4: An example of the bijection between ASM{t,t+1,...,n−1} (n) (left) and monotone triangles with entries ≤ n (right), with t = 4 and n = 6. We can use the following theorem to provide exact formulas for | ASM{t,t+1,...,n−1} (n)| for small values of t, while an exact formula for general t seems out of reach. Theorem 4.3 ([Fis06]). The number of monotone triangles with m rows and prescribed bottom row (k1, k2, … view at source ↗

**Figure 5.** Figure 5: An example of the bijection between ASMI (n) (left) and StI (n) (right), with n = 6 and I = {3}. Acknowledgement We thank Anna Weigandt for helpful conversations and for pointing us towards useful references. References [BB67] B. Banaschewski and G. Bruns. Categorical characterization of the MacNeille completion. Arch. Math. (Basel), 18:369–377, 1967. [BB05] Anders Bj¨orner and Francesco Brenti. Combinator… view at source ↗

read the original abstract

Alternating sign matrices (ASMs) arise as the Dedekind-MacNeille completion of the Bruhat order on the symmetric group. They enjoy fruitful combinatorial and geometric properties, with a particularly rich history on enumerations and bijections. In this paper, we explicitly describe the Dedekind-MacNeille completion of the Bruhat order on any parabolic quotients of the symmetric group. It is naturally a subposet of the alternating sign matrices, with different lattice operations. Moreover, we demonstrate the relations between the meet and join operations in this lattice with taking unions and intersections of the corresponding ASM varieties, respectively. Finally, we conclude with a more detailed discussion of special cases.

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.

Desk Editor's Note private letter to a colleague

Gao and Xu supply an explicit subposet of alternating sign matrices that realizes the Dedekind-MacNeille completion of the Bruhat order on any parabolic quotient W^J, together with a direct link between its meet/join and unions/intersections of the associated ASM varieties.

read the letter

The main contribution is a uniform combinatorial description that extends the known ASM completion for the full symmetric group to arbitrary parabolic quotients. The construction embeds W^J as a join- and meet-dense subposet inside a larger poset of ASMs, equips that poset with its own lattice operations, and proves the whole thing is complete. They also match the new meet and join to unions and intersections of the corresponding ASM varieties by comparing defining ideals directly. That geometric correspondence is the part that feels most useful for people who care about Schubert calculus or partial flag varieties. The special-cases section at the end helps ground the general statement in familiar examples like Grassmannians or two-step flags. The arguments appear to rest on standard order-theoretic facts plus the classical ASM case rather than on any circular definitions, and the stress-test found no gaps in the density or completeness claims. One minor soft spot is that the lattice operations are described as “different” from the usual ASM ones without a side-by-side comparison table; a reader has to reconstruct the distinction from the ideal descriptions. Otherwise the paper is self-contained and the proofs seem to go through for general J. This is aimed at combinatorialists and geometers who already work with Bruhat orders or ASM models and want a single lattice that organizes the parabolic case. It is solid enough to deserve a serious referee; the explicitness and the geometric tie-in make it worth the time even if some readers end up wanting extra examples or computational checks.

Referee Report

0 major / 3 minor

Summary. The paper claims to give an explicit combinatorial description of the Dedekind-MacNeille completion of the Bruhat order on an arbitrary parabolic quotient W^J of the symmetric group S_n. This completion is realized as a subposet of the alternating sign matrices, equipped with modified meet and join operations that differ from the standard ASM lattice operations. The authors further establish that these custom meet and join correspond to unions and intersections of the associated ASM varieties, verify that the quotient embeds as a join- and meet-dense subposet, prove completeness, and conclude with a discussion of special cases.

Significance. If the central claims hold, the work provides a uniform, explicit model for the MacNeille completions of all parabolic quotients of S_n inside the well-studied ASM poset. This extends the classical result for the full symmetric group, supplies a combinatorial handle on these posets, and forges a direct link to the geometry of ASM varieties. The density and completeness arguments, together with the explicit lattice operations, constitute a substantive advance in the order-theoretic and geometric combinatorics of Coxeter groups and their quotients.

minor comments (3)

[Abstract] Abstract: the phrase 'with different lattice operations' is accurate but vague; a single sentence indicating how the new meet and join are defined (e.g., via a combinatorial rule on the ASM entries) would improve immediate readability.
[Introduction] The manuscript refers to 'the corresponding ASM varieties' without a preliminary definition or reference in the opening paragraphs; a short paragraph recalling the ideal-theoretic or matrix-variety construction would help readers who are not already expert in the geometric side.
[Final section] In the discussion of special cases, the authors could usefully include a small table or explicit listing for n=4 or n=5 and representative J (e.g., J={s_2}) showing the added elements and the new meet/join tables; this would make the general construction more concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the accurate summary of our main results on the MacNeille completion of parabolic quotients and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends the established result that alternating sign matrices realize the Dedekind-MacNeille completion of the Bruhat order on the full symmetric group (a result independent of the present authors) to arbitrary parabolic quotients W^J. It supplies an explicit combinatorial subposet of ASMs, proves join- and meet-density of the quotient inside this subposet, establishes completeness, and verifies that the induced meet/join operations coincide with unions/intersections of the associated ASM varieties by direct comparison of defining ideals. No step reduces by construction to a self-definition, a fitted input renamed as a prediction, or a load-bearing self-citation chain; the argument relies on standard order-theoretic definitions and explicit constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definition of Dedekind-MacNeille completion and the known fact that ASMs complete the Bruhat order on the full symmetric group; no free parameters or new entities are introduced.

axioms (2)

standard math The Dedekind-MacNeille completion of any poset is the smallest lattice containing it.
Standard theorem in order theory invoked implicitly throughout.
domain assumption Alternating sign matrices form the Dedekind-MacNeille completion of the Bruhat order on the symmetric group.
Cited as background in the abstract.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Algebra and geometry of ASM weak order.arXiv preprint arXiv:2502.19266,

[EKW25] Laura Escobar, Patricia Klein, and Anna Weigandt. Algebra and geometry of ASM weak order.arXiv preprint arXiv:2502.19266,

work page arXiv
[2]

[Ste96] John R

With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. [Ste96] John R. Stembridge. On the fully commutative elements of Coxeter groups.J. Algebraic Combin., 5(4):353–385,

1986