Rowmotion and Echelonmotion

Adrien Segovia; Colin Defant; David E Speyer; Hugh Thomas; Nathan Williams; Rene Marczinzik; Yuhan Jiang

Rowmotion and Echelonmotion

T0 review · 0 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Echelonmotion and rowmotion are the same bijection on every semidistributive lattice, and the converse characterizes echelon-independence.

desk verdict Genuine generalization of KMT25: echelonmotion equals rowmotion on semidistributive lattices, with a solid trim-lattice analogue and a clean Eulerian involution result; the main external dependency checks out. read the letter →

arxiv 2507.18230 v1 pith:VYOATWSO submitted 2025-07-24 math.CO

Colin Defant , Yuhan Jiang , Rene Marczinzik , Adrien Segovia , David E Speyer , Hugh Thomas , Nathan Williams This is my paper · ORCID

classification math.CO MSC 06A0706B0515A23

keywords echelonmotionrowmotionsemidistributivelatticestrimEulerianposetsBruhatdecompositionechelon-independencelinearextensions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Echelonmotion is a matrix-factorization recipe that assigns to a finite poset and a linear extension a bijection of the poset. The paper proves that on semidistributive lattices this bijection is exactly rowmotion, the standard dynamical map that had previously been understood mainly on distributive lattices. It further proves the converse: a lattice is echelon-independent if and only if it is semidistributive. For trim lattices, which need not be semidistributive, the paper identifies a class of vertebral linear extensions for which the equality with rowmotion still holds, and it shows that on Eulerian posets echelonmotion is an involution. The paper thereby gives rowmotion a uniform matrix realization on much broader poset families and an efficient test for when that realization is canonical.

What carries the argument

The central object is echelonmotion: for an $n$-element poset $R$ and a linear extension $\sigma$, form the $n \times n$ Cartan matrix $W_{R,\sigma}$, factor it under the double-coset decomposition $\mathrm{GL}_n(\mathbb{C}) = \bigsqcup_{P \in S_n} BPB$, and let the unique permutation matrix define the bijection $\mathrm{Ech}_\sigma \colon R \to R$. The key mechanism is Proposition 3.3: if a lattice $L$ satisfies $\mu_L(\mathrm{Pop}_L(x), x) \neq 0$ for every $x$ and the map $x \mapsto \max_\sigma(\Upsilon_L(x))$ is a bijection, then $\mathrm{Ech}_\sigma(x) = \max_\sigma(\Upsilon_L(x))$, where $\Upsilon_L(x)$ is the set of elements whose meet with $x$ equals $\mathrm{Pop}_L(x)$. A known characterization of rowmotion identifies this maximum with $\mathrm{Row}_L(x)$ on semidistributive lattices, so the proposition converts echelonmotion into rowmotion; the same proposition, applied with vertebral linear extensions, handles trim lattices, and a separate argument shows the permutation matrix is its own inverse on Eulerian posets.

What would settle it

Find a finite lattice that is echelon-independent but not semidistributive, which would refute the converse half of Theorem 1.3; more directly, compute $\mu_L(\mathrm{Pop}_L(x), x)$ on any semidistributive or trim lattice and look for a zero, or compare $\mathrm{Ech}_\sigma(x)$ with $\mathrm{Row}_L(x)$ across all linear extensions and elements of a single lattice.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.3: if $L$ is a finite semidistributive lattice, then $\mathrm{Ech}_\sigma = \mathrm{Row}_L$ for every linear extension $\sigma$ of $L$, where $\mathrm{Row}_L$ is the rowmotion operator for semidistributive lattices. In particular, every semidistributive lattice is echelon-independent, meaning the chosen linear extension is irrelevant. The converse is also proved: every echelon-independent lattice is semidistributive. For trim lattices, the paper shows that a specially chosen vertebral linear extension still makes $\mathrm{Ech}_\sigma$ agree with rowmotion, and for Eulerian posets it proves that $\mathrm{Ech}_\sigma$ is an involution for every linear extension.

Load-bearing premise

The proof that echelonmotion equals rowmotion rests on the imported fact that in every semidistrim lattice the Möbius value $\mu_L(\mathrm{Pop}_L(x), x)$ is never zero; if some semidistributive or trim lattice violated that fact, the identification would not follow from Proposition 3.3.

Editorial extensions

If this is right

Every semidistributive lattice is echelon-independent, so its rowmotion bijection can be obtained from any linear extension by a single double-coset factorization of its Cartan matrix.
An echelon-independent connected poset is bounded, its echelonmotion is fixed-point-free when the poset has at least two elements, and its MacNeille completion is semidistributive.
Trim lattices that are not semidistributive still have a vertebral linear extension on which echelonmotion and rowmotion coincide.
Eulerian posets, including face lattices of polytopes and intervals in the strong Bruhat order, carry an echelonmotion involution for every linear extension.
Echelon-independence can be tested by computing ranks of at most $16n$ matrices rather than enumerating all linear extensions of an $n$-element poset.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the paper's rank-based test could be used as a fast certificate that a lattice is not semidistributive: a single pair of linear extensions with different echelonmotion would refute semidistributivity without constructing the full canonical join complex.
Editorial inference: the vertebral-linear-extension construction is a natural candidate to transfer from trim lattices to independence posets, in which case the equality between echelonmotion and rowmotion would give that larger class a canonical matrix realization.
Editorial inference: because echelonmotion is defined for any finite-dimensional algebra with invertible Cartan matrix, the lattice-theoretic results point toward an algebraic analogue of rowmotion whose incidence-algebra special case is the poset echelonmotion studied here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a genuine generalization of KMT25: echelonmotion, defined via the Bruhat decomposition of Cartan matrices, is shown to coincide with Barnard's rowmotion on every semidistributive lattice, independent of the chosen linear extension. That is the main result, and it is new. The trim-lattice theorem is also new: for every trim lattice there is a specially constructed 'vertebral' linear extension making echelonmotion agree with rowmotion. The Eulerian involution theorem is a short, elegant linear-algebra argument, and the structural results on echelon-independent posets (bounded, no fixed points, semidistributive MacNeille completion) are useful.

The proofs are carefully constructed. Proposition 3.3 is a clean reduction: if the Möbius values mu(Pop(x), x) are nonzero and x maps to max_sigma Upsilon(x) bijectively, then Ech_sigma equals that map. In the semidistributive case, equation (5) identifies that maximum with Row(x), so the result follows. The trim case is a longer induction, but the logic is visible and the use of [DW23, Proposition 9.5] and Lemma 6.3 is sound. The counterexample to the converse of Theorem 1.8 is explicit and convincing.

The main dependency is Lemma 2.3, imported from Defant--Williams, asserting mu(Pop(x), x) is +/-1 for semidistrim lattices. If that failed, the central equivalence would collapse. I checked the derivation: for Q = [Pop(x), x], the interval's maximum is Pop(x), the interval's minimum, so the contractibility condition in [DW23, Corollary 8.2] fails, forcing the order complex to be sphere-like and the Möbius value nonzero. The dependency is solid. The other external input, equation (5), is also an established published result.

The only soft spot worth naming is that the vertebral linear extension construction in Section 6 is somewhat heavy, and the proof there is intricate; an independent check of the induction would be wise. But the paper is honest about what is imported, the computational checks are clearly labeled as checks, and I found no circularity or fitting. The citation pattern is reasonable: prior work by the same group is cited for exactly the needed lemmas, and those lemmas are published.

This paper deserves a serious referee. It is not a desk reject. I would cite it in future work on rowmotion and lattice dynamics, and I would bring it to reading group.

Referee Report

0 major / 3 minor

Summary. The paper introduces echelonmotion, a bijection Ech_σ on a finite poset R defined from the Bruhat decomposition of the Cartan matrix of R with respect to a linear extension σ. It proves that on semidistributive lattices this bijection coincides with rowmotion for every linear extension (Theorem 1.3), and that a lattice is echelon-independent exactly when it is semidistributive. For trim lattices, the authors define vertebral linear extensions and prove that echelonmotion with respect to any such extension equals rowmotion (Theorem 1.4). It further shows that echelonmotion on Eulerian posets is an involution (Theorem 1.5), and gives several structural results for echelon-independent connected posets, including boundedness (Theorem 1.6), absence of fixed points (Theorem 1.7), and semidistributivity of the MacNeille completion (Theorem 1.8). An efficient algorithm for testing echelon-independence is presented and used to report that the Bruhat order on S_n is echelon-independent for n ≤ 5 but not for n = 6.

Significance. The paper substantially generalizes the distributive-lattice result of Klász–Marczinzik–Thomas and connects rowmotion to a concrete linear-algebraic construction via Bruhat decomposition. The characterization of echelon-independent lattices as semidistributive is clean and nontrivial, and the trim-lattice result provides a new instance where echelonmotion matches rowmotion even outside the semidistributive setting. The proofs are systematic and transparent, using the rank criterion (3) and the pop-stack operator. A particular strength is that the main external dependencies, especially Lemma 2.3 imported from Defant–Williams [DW23] and equation (5), are published results that do not assume the target equality; I checked the derivation of Lemma 2.3 from [DW23, Corollary 8.2 and Theorem 7.8] and it is valid. The paper also provides concrete computational evidence, including the S_6 counterexample and tests for modular lattices, making the claims easy to verify.

minor comments (3)

[Section 3, proof of Proposition 3.3] In the displayed equation after 'Since u ∧ x ≠ Pop_L(x)', the summation set is written as Δ_Qx(u ∧ v), but v is not defined; it should be Δ_Qx(u ∧ x).
[Section 4, proof of Theorem 1.8] After invoking [Sch16, Proposition 8.26], the assertion 'This implies that J_L and M_L are contained in R' would be clearer with a one-sentence justification: a join-irreducible element that is a join of elements of R must be one of those elements, and dually for meet-irreducibles.
[Section 2, Lemma 2.3] The proof of Lemma 2.3 is only sketched through a chain of citations to [DW23]. Since this lemma is load-bearing for Theorems 1.3 and 1.4, a slightly fuller explanation of the interval reduction to [DW23, Corollary 8.2] would improve readability, even though the cited result is published and correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the echelonmotion-rowmotion equality is proved from independent cited results, not built into the definitions.

full rationale

The main theorem (Theorem 1.3) is not circular. Ech_sigma is defined from the Bruhat cell of the Cartan matrix (Definition 1.1), entirely without reference to rowmotion; Row_L is defined by the Barnard characterization UL(Row_L(w)) = DL(w). The bridge, Proposition 3.3, proves Ech_sigma(x) = max_sigma(Ypsilon_L(x)) whenever that map is a bijection and mu_L(Pop_L(x),x) is nonzero. The two inputs are external: Lemma 2.3 (from Defant-Williams, DW23) proves the Möbius nonvanishing for semidistrim lattices, and equation (5) (also DW23) identifies max Ypsilon(x) with Row_L(x). Neither input mentions echelonmotion or assumes Ech = Row, so invoking them is independent support rather than a self-citation smuggling operation. The trim-lattice theorem has the same structure: it proves max_sigma Ypsilon = Row by induction and feeds that into Proposition 3.3. The converse implication in Theorem 1.3 follows from Theorem 1.8, whose proof uses Proposition 2.2 and Lemma 2.1 to construct two linear extensions with different echelonmotion images; it does not assume that an echelon-independent lattice is already semidistributive. Although several cited results (DW23, KMT25, TW19b) share authors with the present paper, none of those citations has as its conclusion the target equality Ech = Row; the cited statements have independent stated assumptions that do not include the target result. There is no fitted parameter renamed as a prediction, no equation that reduces to its own input by construction, and no uniqueness assertion imported to force the choice. Therefore the derivation chain is self-contained with respect to external notions and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

Pure mathematics: no free parameters fitted to data. The central claims rest on standard linear algebra (Bruhat decomposition and rank criteria) and on several prior lattice-theoretic results, some authored by members of this paper's group, which are used as background. The new definitions (echelonmotion, vertebral linear extensions, echelon-independent posets) are mathematical objects with no external empirical content.

assumptions (5)

standard math Bruhat decomposition of GL_n(C) and the rank criterion (3) that locates the permutation matrix P_{R,σ}.
Used throughout Section 3 and in Propositions 3.1-3.3 to characterize echelonmotion.
domain assumption Lemma 2.3: for a semidistrim lattice L, μ_L(Pop_L(x), x) ∈ {−1,1}, imported from [DW23].
Load-bearing for Proposition 3.3 and hence for Theorems 1.3 and 1.4; it is cited, not proved in this paper.
domain assumption Equation (5): for a semidistributive lattice, max Υ_L(x) = {Row_L(x)}, imported from [DW23].
Used in proof of Theorem 1.3 to identify echelonmotion with rowmotion.
domain assumption Equation (7): Row_L(x) ∈ max Υ_L(x) for trim lattices, imported from [DW23, Theorem 9.3].
Used in the trim-lattice proof of Theorem 1.4.
domain assumption MacNeille completion properties, in particular [Sch16, Proposition 8.26] that every element of the MacNeille completion of R is a join and meet of elements of R.
Foundational for Theorem 1.8 and the containment JL, ML ⊆ R.

invented entities (2)

Echelonmotion Ech_σ
purpose: Bijection R→R defined via Bruhat decomposition of the Cartan matrix; the central object of study.
New mathematical definition; no empirical prediction. Consistency is established by the theorems rather than external evidence.
Vertebral linear extension
purpose: A linear extension σ_C of a trim lattice constructed from a maximum-length chain C, for which echelonmotion is proven to equal rowmotion.
New construct introduced in Definition 6.7.

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Cite this review

Pith. "Pith review of Rowmotion and Echelonmotion." pith.science (2026). https://pith.science/paper/VYOATWSO

@misc{pith2026250718230,
  author       = {Pith},
  title        = {Pith review of: Rowmotion and Echelonmotion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYOATWSO}},
  note         = {Machine review of arXiv:2507.18230}
}

abstract

Given a linear extension $\sigma$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $\sigma$. This yields a bijection $\mathrm{Ech}_\sigma\colon R\to R$ that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Kl\'asz, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when $R$ is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.

Figures

Figures reproduced from arXiv: 2507.18230 by the authors.

**Figure 1.** On the left is the distributive lattice of lower order ideals of a 3-element poset, where each lower order ideal is represented as a collection of elements shaded in red. On the right is the same lattice labeled according to a linear extension. In each depiction of the lattice, a green arrow is drawn from an element to its image under rowmotion. Example 1.2. Let Q be the 3-element poset . The left side of [PITH_FUL… view at source ↗

**Figure 2.** A lattice L such that PopL(ˆ1) = ˆ0 and µ(ˆ0, ˆ1) = 0. Lemma 4.1. Let R be a poset of size n. Let x be a minimal element of R, and let y be a maximal element of R such that x ≤ y. There exists a linear extension σ of R such that σ(x) = 1 and σ(y) = n. Moreover, for any such linear extension σ, we have Echσ(x) = y. Proof. The existence of σ is straightforward. We have Preσ(x) = {x}, and Sucσ(y) = {y}. Define ϱ: Preσ(… view at source ↗

**Figure 3.** Two linear extensions of a poset whose MacNeille completion is distributive. Echelonmotion with respect to each linear extension is represented by green arrows; note that the two echelonmotion maps are different. Proposition 4.2. Let σ # be a linear extension of a poset R. Fix x ∈ R, and let y = Echσ# (x). Suppose x and y are comparable. Fix λ1 ∈ Λ1(x, y) and λ2 ∈ Λ2(x, y). If Echλ1 (x) = Echλ2 (x) = y, then Echσ(x… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A semidistributive lattice. Each join-irreducible element has its own color, which is also used to color the edges that it labels. A green arrow is drawn from each element to its image under rowmotion. Proof of Theorem 1.3. Fix a linear extension σ of a semidistributiv…

**Figure 5.** Figure 5: On the left is a trim lattice with the elements of a maximum-length chain C = {u0, u1, u2, u3, u4} labeled. Each element x is represented by a circle filled with σC(x). We have also labeled the join-irreducible elements j1, j2, j3, j4 and the meet-irreducible elements …

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A survey on Auslander-Gorenstein algebras
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A survey of finite-dimensional Auslander-Gorenstein algebras, including classifications for monomial and incidence algebras and the identification of the Auslander-Reiten permutation with rowmotion, Ringel's homologic...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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