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arxiv: 2605.25727 · v1 · pith:BDI64X2Dnew · submitted 2026-05-25 · 🧮 math.CO

A Bruhat order for Latin squares and alternating sign hypermatrices

Angela Carnevale , Cian O'Brien This is my paper

Pith reviewed 2026-06-29 21:42 UTC · model grok-4.3

classification 🧮 math.CO
keywords Bruhat orderLatin squaresalternating sign hypermatricescorner-sum hypermatricesdistributive latticeDedekind-MacNeille completionmonotone hypertriangles
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The pith

Entrywise domination on corner-sum hypermatrices encodes a Bruhat order on Latin squares and alternating sign hypermatrices.

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

The paper extends the classical Bruhat order on permutation matrices and alternating sign matrices to the three-dimensional setting of Latin squares and alternating sign hypermatrices. It introduces corner-sum hypermatrices C_n and proves that the order is realized precisely by entrywise comparison of these arrays. The resulting poset on C_n forms a distributive lattice, yet it fails to be the Dedekind-MacNeille completion of the poset of Latin squares. The authors further characterize covering relations, give rank formulas, and exhibit a bijection with monotone hypertriangles that also encodes the order by entrywise domination.

Core claim

We define a Bruhat order ≼_B on Latin squares and alternating sign hypermatrices by entrywise domination on the associated corner-sum hypermatrices C_n; C_n is a distributive lattice but is not the Dedekind-MacNeille completion of the Latin-square poset, and monotone hypertriangles stand in bijection with C_n while likewise encoding the order.

What carries the argument

The corner-sum hypermatrix C_n, whose (i,j,k) entry is the sum of all entries of the input hypermatrix with indices at most (i,j,k); entrywise domination on these arrays defines the order ≼_B.

If this is right

  • Covering relations in the poset of C_n admit an explicit combinatorial description.
  • Rank functions on C_n generalize the known formulas for alternating sign matrices.
  • Monotone hypertriangles provide an equivalent encoding of the order via entrywise domination.
  • The order on alternating sign hypermatrices extends the order on Latin squares in a manner directly analogous to the two-dimensional case.

Where Pith is reading between the lines

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

  • The failure of C_n to complete the Latin-square poset suggests that higher-dimensional Bruhat orders may require additional elements or a modified completion construction.
  • The bijection with monotone hypertriangles offers a potential route to study the order through simpler triangular arrays rather than full hypermatrices.
  • Rank formulas may allow enumeration or probabilistic analysis of chains in the poset of Latin squares under this order.

Load-bearing premise

The three-dimensional corner-sum construction is the correct direct analogue of the two-dimensional corner-sum matrix for encoding the Bruhat order.

What would settle it

A pair of Latin squares (or alternating sign hypermatrices) in which one entrywise-dominates the other on C_n yet the proposed Bruhat relation ≼_B fails to hold between them, or vice versa.

Figures

Figures reproduced from arXiv: 2605.25727 by Angela Carnevale, Cian O'Brien.

Figure 1
Figure 1. Figure 1: Hasse diagram of the Bruhat order on 3 × 3 permutation matrices. For all n ∈ N, the identity matrix In is the unique minimal element of the poset of permutation matrices under the Bruhat order, and the complete inversion Jn of In is the unique maximal element. For n ⩾ 3, however, n × n permutation matrices under the Bruhat order do not form a lattice. 1.2 Alternating sign matrices and corner-sum matrices T… view at source ↗
Figure 2
Figure 2. Figure 2: The Hasse diagrams for the ASMs (left) and corner-sum matrices (right) of order 3 under [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Hasse diagram of L3. In [22], it is shown that for arbitrarily large n ∈ N, there exist pairs of Latin squares of order n which cannot be obtained from one another by a sequence of cycle switches. Cycle switches, however, are useful to describe the Bruhat relation for Latin squares, and to relate it to the 2-dimensional Bruhat order. Theorem 2.7. Let L1, L2 ∈ Ln be such that L1 can be obtained from L2 … view at source ↗
Figure 4
Figure 4. Figure 4: The Hasse diagram of the Bruhat order on ASHMs of order 3. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Hasse diagram of C3. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
read the original abstract

The Bruhat order on permutation matrices extends to alternating sign matrices via corner-sum matrices, where the order is given by entrywise domination. A classical result of Lascoux and Sch\"utzenberger states that alternating sign matrices form the Dedekind-MacNeille completion of the Bruhat order on permutations. Brualdi and Dahl introduced alternating sign hypermatrices as a three-dimensional analogue of alternating sign matrices and used them to generalise Latin squares, which may be viewed as three-dimensional analogues of permutation matrices. In this paper, in analogy with the two-dimensional case, we define and study a Bruhat order $\preceq_B$ on Latin squares and alternating sign hypermatrices. We introduce the corresponding corner-sum hypermatrices $\mathcal C_n$ and prove that entrywise domination on $\mathcal C_n$ encodes this order. We show that $\mathcal C_n$ is a distributive lattice, but that, unlike in dimension two, it is not the Dedekind-MacNeille completion of the poset of Latin squares. We further characterise the covering relations for $\mathcal C_n$ and prove rank formulae generalising the classical case of alternating sign matrices. Finally, we define monotone hypertriangles, prove that they are in bijection with $\mathcal C_n$, and show that they also encode the order by entrywise domination.

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

0 major / 3 minor

Summary. The paper extends the Bruhat order from permutation matrices and alternating sign matrices to the three-dimensional setting of Latin squares and alternating sign hypermatrices. It defines ≼_B via entrywise domination on the associated corner-sum hypermatrices C_n (in direct analogy with the 2D case), proves that this encodes the order, shows that C_n is a distributive lattice but not the Dedekind-MacNeille completion of the poset of Latin squares, characterizes covering relations and rank functions generalizing the ASM case, and establishes a bijection with monotone hypertriangles that likewise encode the order by domination.

Significance. If the stated theorems hold, the work supplies a coherent combinatorial generalization of the Bruhat order and corner-sum construction to hypermatrices, with explicit lattice-theoretic properties, covering relations, and rank formulae. The explicit demonstration that C_n is not the Dedekind-MacNeille completion (in contrast to the 2D Lascoux-Schützenberger theorem) and the bijection with monotone hypertriangles are useful distinctions and alternative models. The parameter-free, definition-driven approach and the provision of multiple equivalent encodings strengthen the contribution to poset combinatorics.

minor comments (3)
  1. [§3] §3, Definition 3.2: the recursive definition of the corner-sum hypermatrix C_n would be clearer if accompanied by an explicit computation for n=2 or n=3, showing how the entries are obtained from a Latin square or ASHM.
  2. [§5] §5, Theorem 5.4: the statement that C_n is not the Dedekind-MacNeille completion would benefit from a brief indication of the minimal missing elements or a reference to the specific pair of Latin squares whose join is absent.
  3. [§7] The notation for monotone hypertriangles in §7 occasionally re-uses symbols already employed for hypermatrices; a short notational table or remark distinguishing the two would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; new definitions with independent proofs

full rationale

The paper defines ≼_B on Latin squares and alternating sign hypermatrices explicitly via entrywise domination on the newly introduced corner-sum hypermatrices C_n, stated as an analogy to the 2D case. It then proves independent properties including that this encodes the order, that C_n is a distributive lattice but not the Dedekind-MacNeille completion of the Latin square poset, covering relations, rank formulae, and a bijection with monotone hypertriangles. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; all central claims are new constructions followed by direct proofs without load-bearing self-references or renamings of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Relies on standard facts about posets, distributive lattices, and the classical two-dimensional Bruhat order; introduces new objects without free parameters or external data fits.

axioms (2)
  • standard math The Bruhat order on permutation matrices is given by entrywise domination of their corner-sum matrices (Lascoux-Schützenberger).
    Invoked as the base case for the three-dimensional analogy.
  • domain assumption Alternating sign hypermatrices and Latin squares are the appropriate three-dimensional analogues of alternating sign matrices and permutation matrices (Brualdi-Dahl).
    Used to motivate the objects on which the new order is defined.
invented entities (2)
  • corner-sum hypermatrices C_n no independent evidence
    purpose: Encode the Bruhat order on Latin squares and alternating sign hypermatrices via entrywise domination.
    Newly defined in the paper; no independent evidence supplied outside the construction.
  • monotone hypertriangles no independent evidence
    purpose: Objects in bijection with C_n that also encode the order by entrywise domination.
    Newly introduced; bijection and order-encoding are proved within the paper.

pith-pipeline@v0.9.1-grok · 5774 in / 1355 out tokens · 28423 ms · 2026-06-29T21:42:59.852262+00:00 · methodology

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Reference graph

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