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arxiv: 2606.13518 · v1 · pith:6R77W4AUnew · submitted 2026-06-11 · 🧮 math.CO

Weak order: Alternating sign matrices, monotone triangles, and bumpless pipe dreams

Laura Escobar , Patricia Klein , Anna Weigandt This is my paper

Pith reviewed 2026-06-27 06:15 UTC · model grok-4.3

classification 🧮 math.CO
keywords alternating sign matricesweak ordermonotone trianglesbumpless pipe dreamsBruhat ordercovering relationscombinatorial bijections
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The pith

Two definitions of weak order on alternating sign matrices coincide and admit explicit covering rules in three models.

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

The paper reconciles an order on alternating sign matrices coming from monotone triangles with an earlier direct definition used for geometric purposes. Establishing that the two orders agree lets combinatorial and algebraic results move freely between the models. The authors supply concrete rules for deciding when one matrix covers another, phrased directly on the matrices, on the triangles (by a method different from earlier work), and on bumpless pipe dreams. In the pipe-dream language they further show that the sets of matrices mapping to the same element under the weak-order operators form sublattices inside the stronger Bruhat order.

Core claim

The weak order on ASM(n) induced from monotone triangles via the standard bijection is identical to the a priori different weak order previously defined directly on ASMs. Explicit covering relations are given on ASMs themselves, on monotone triangles by a rule distinct from Hamaker-Reiner, and on bumpless pipe dreams. In the bumpless-pipe-dream model the fibers of the weak-order operators are each a sublattice of the strong Bruhat order on ASM(n).

What carries the argument

The bijections among alternating sign matrices, monotone triangles, and bumpless pipe dreams that preserve weak-order covering relations.

If this is right

  • Covering relations can be read off directly from the pattern of entries in an alternating sign matrix.
  • A distinct rule computes covers when the same matrices are viewed as monotone triangles.
  • Bumpless pipe dreams supply a third explicit combinatorial rule for the covers.
  • Each fiber of a weak-order operator is a sublattice inside the Bruhat order when realized by bumpless pipe dreams.

Where Pith is reading between the lines

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

  • The compatibility may let geometric or K-theoretic statements proved in one model be restated combinatorially in another.
  • The sublattice property on fibers could be used to study chain decompositions or representation-theoretic multiplicities inside the Bruhat order.
  • Analogous compatibility statements might hold for other partial orders defined on the same three families of objects.

Load-bearing premise

The standard bijections between alternating sign matrices, monotone triangles, and bumpless pipe dreams preserve the covering relations of the two independently defined weak orders.

What would settle it

A concrete pair of n by n alternating sign matrices that cover each other under one definition of weak order but fail to cover under the other.

read the original abstract

In 2018, Hamaker and Reiner introduced weak order for monotone triangles, which extended the usual notion of weak order on the symmetric group. Monotone triangles on $\{1, \ldots, n\}$ are well-known to be in bijection with the set ASM$(n)$ of $n \times n$ alternating sign matrices. Hamaker and Reiner defined weak order on ASM$(n)$ to be induced from weak order on monotone triangles via the standard bijection. Recently, the present authors used an a priori different definition of weak order on ASM$(n)$ to give a combinatorial characterization of the codimension of ASM varieties and to show that the natural K-theoretic representatives of these varieties satisfy a divided difference recurrence. In the present work, we establish compatibility of these definitions of weak order on ASM$(n)$. Additionally, we give three different explicit means of computing weak order covering relations on ASM$(n)$: on ASMs themselves, on monotone triangles in a manner different from that given by Hamaker and Reiner, and on bumpless pipe dreams, which are a newer family of combinatorial objects also in correspondence with ASMs. Finally, using the language of bumpless pipe dreams, we characterize the fibers of the weak order operators, each of which forms a sublattice of the strong Bruhat order on ASM$(n)$.

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 establishes compatibility between an a priori definition of weak order on ASM(n) and the definition induced from Hamaker-Reiner weak order on monotone triangles via the standard bijection. It supplies three explicit combinatorial rules for computing covering relations (directly on ASMs, on monotone triangles in a manner distinct from Hamaker-Reiner, and on bumpless pipe dreams) and, in the bumpless-pipe-dream model, characterizes the fibers of the weak-order operators as sublattices of the strong Bruhat order on ASM(n).

Significance. If the compatibility statement holds, the work unifies previously separate combinatorial models of weak order on alternating sign matrices and supplies practical, explicit covering rules that can be used for computation. The fiber characterization in the bumpless-pipe-dream language is a concrete combinatorial contribution that strengthens the link to the authors' earlier results on ASM varieties and their K-theoretic representatives. The explicitness of the three covering rules is a strength of the manuscript.

major comments (2)
  1. [§3] The central compatibility claim requires that the three covering rules coincide under the known bijections between ASMs, monotone triangles, and bumpless pipe dreams. The manuscript states the rules and asserts compatibility, but the load-bearing verification that each pair of rules is transported identically by the bijections (for arbitrary n and arbitrary pairs of objects) is not carried out in sufficient detail; a mismatch on even one pair would falsify the claim that the a priori ASM order coincides with the induced order.
  2. [§5] §5 (fiber characterization): the proof that each fiber is a sublattice of the strong Bruhat order relies on the covering rules being compatible with the bijections; without an explicit check that the bumpless-pipe-dream covering rule is the image of the ASM covering rule, the sublattice statement remains conditional on the unverified transport.
minor comments (2)
  1. Notation for the three covering relations is introduced separately; a single comparative table or diagram showing the three rules side-by-side would improve readability.
  2. [§2] The manuscript refers to 'the standard bijection' between ASMs and monotone triangles without restating its definition; including a brief recall (or reference to a numbered equation) would make the transport arguments self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the compatibility arguments. We respond to the major comments below.

read point-by-point responses

  1. Referee: [§3] The central compatibility claim requires that the three covering rules coincide under the known bijections between ASMs, monotone triangles, and bumpless pipe dreams. The manuscript states the rules and asserts compatibility, but the load-bearing verification that each pair of rules is transported identically by the bijections (for arbitrary n and arbitrary pairs of objects) is not carried out in sufficient detail; a mismatch on even one pair would falsify the claim that the a priori ASM order coincides with the induced order.

    Authors: We agree that the verification of compatibility under the bijections would benefit from a more explicit and self-contained presentation. In the revised manuscript we will insert a dedicated subsection (or short appendix) that records the image of each covering rule under the standard bijections, with a general outline for arbitrary n together with illustrative diagrams for small n to confirm that the three rules are transported identically. revision: yes

  2. Referee: [§5] §5 (fiber characterization): the proof that each fiber is a sublattice of the strong Bruhat order relies on the covering rules being compatible with the bijections; without an explicit check that the bumpless-pipe-dream covering rule is the image of the ASM covering rule, the sublattice statement remains conditional on the unverified transport.

    Authors: The argument in §5 is indeed conditional on the compatibility of the covering rules. Once the enhanced verification described in our response to the preceding comment is added, the sublattice claim follows directly. We will also add a short clarifying sentence in §5 that explicitly references the updated compatibility argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; compatibility and explicit rules are derived independently using external bijections as input.

full rationale

The paper takes the standard ASM-monotone triangle bijection and the Hamaker-Reiner weak order on triangles as given external facts. It introduces an a priori definition of weak order on ASMs from the authors' prior work, then proves compatibility by supplying three new explicit covering-relation rules (on ASMs, on triangles, and on bumpless pipe dreams) and verifying transport under the known bijections. No equation or claim reduces a derived quantity to a fitted parameter or to a self-referential definition; the central result is the verification itself, which is not assumed or imported via self-citation. Prior self-citation is limited to the source of one of the two orders being compared and is not load-bearing for the compatibility statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard bijections between ASMs, monotone triangles, and bumpless pipe dreams that are taken from the existing literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption There exists a standard bijection between monotone triangles on {1,...,n} and n x n alternating sign matrices that induces the weak order.
    Invoked to transfer the 2018 Hamaker-Reiner definition to ASMs.
  • domain assumption Bumpless pipe dreams are in bijection with alternating sign matrices.
    Invoked to obtain the third explicit means of computing covering relations.

pith-pipeline@v0.9.1-grok · 5773 in / 1503 out tokens · 23577 ms · 2026-06-27T06:15:08.102681+00:00 · methodology

discussion (0)

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Reference graph

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