Echelonmotion in Finite Posets

• Echelonmotion is a bijective map on finite posets defined via Cartan matrices and Bruhat decomposition that exactly recovers classical rowmotion.
• It equals rowmotion on semidistributive and vertebrally ordered trim lattices, underpinning results on σ-independence and lattice symmetry.
• The theory reveals an involutive structure on Eulerian posets and provides new algebraic proofs of cover statistic equalities in modular lattices.

Searching arXiv for the cited papers and related work on echelonmotion to ground the article. Echelonmotion is a bijection attached to a finite poset together with a chosen linear extension. In the formulation of Defant–Jiang–Marczinzik–Segovia–Speyer–Thomas–Williams, one forms a Cartan matrix from the order relation, locates the unique Bruhat cell of $GL_n(\mathbb{C})$ containing that matrix, and reads the associated permutation matrix back as a permutation of the poset. The resulting map, denoted $\mathrm{Ech}_\sigma$, is the inverse of the Coxeter permutation studied by Klász, Marczinzik, and Thomas. Its central significance is that it recovers rowmotion in broad lattice-theoretic settings, exhibits involutive behavior on Eulerian posets, and supports structural characterizations of $\sigma$-independence and new results in modular lattice theory (Defant et al., 24 Jul 2025).

1. Definition through Cartan matrices and Bruhat decomposition

Let $R$ be a finite poset of size $n$, and fix a linear extension $\sigma\colon R\to[n]$, meaning that $\sigma$ is a bijection with $\sigma(x)\le \sigma(y)$ whenever $x\le y$ in $R$. The Cartan matrix $\mathrm{Ech}_\sigma$0 is defined by

$\mathrm{Ech}_\sigma$1

Because $\mathrm{Ech}_\sigma$2 is lower-triangular with all diagonal entries $\mathrm{Ech}_\sigma$3, it is invertible and lies in the big cell of the Bruhat decomposition

$\mathrm{Ech}_\sigma$4

where $\mathrm{Ech}_\sigma$5 is the subgroup of invertible upper-triangular matrices and each $\mathrm{Ech}_\sigma$6 is an $\mathrm{Ech}_\sigma$7 permutation matrix. Hence there is a unique permutation matrix $\mathrm{Ech}_\sigma$8 such that $\mathrm{Ech}_\sigma$9 (Defant et al., 24 Jul 2025).

Echelonmotion with respect to $\sigma$0 is then the bijection $\sigma$1 determined by

$\sigma$2

Equivalently, if $\sigma$3 with $\sigma$4 and $\sigma$5 upper-triangular and $\sigma$6, then $\sigma$7 is the permutation underlying that factorization after relabeling by $\sigma$8. The permutation matrix can also be recovered by rank conditions on lower-left submatrices: for $\sigma$9, one has $R$0 if and only if

$R$1

An explicit example appears for the $R$2-element distributive lattice $R$3 of a $R$4-element poset, where the paper writes

$R$5

and the induced permutation agrees with classical rowmotion.

2. Agreement with rowmotion on semidistributive lattices

The primary structural theorem identifies echelonmotion with rowmotion on semidistributive lattices. Recall that a lattice $R$6 is meet-semidistributive if whenever $R$7 the set $R$8 has a greatest element; join-semidistributive is defined dually; and semidistributive means both hold. Barnard’s rowmotion on a semidistributive lattice may be defined via the canonical edge-labelling by join-irreducibles. Equivalently, using the pop-stack operator

$R$9

one has

$n$0

Defant–Williams showed the equivalence of these descriptions, and Defant–Jiang–Marczinzik–Segovia–Speyer–Thomas–Williams used them to prove that for a semidistributive lattice $n$1 and any linear extension $n$2,

$n$3

In particular, $n$4 is echelon-independent (Defant et al., 24 Jul 2025).

The proof strategy depends on the interaction between Bruhat rank tests and Möbius-theoretic information. In any semidistrim lattice, hence in particular in any semidistributive lattice, the Möbius function $n$5 is always $n$6, so the interval $n$7 contributes a nonzero coefficient in the Bruhat-rank tests. Writing $n$8, Corollary 3.6 in the paper shows that whenever $n$9 and $\sigma\colon R\to[n]$0 is a bijection, the rank tests force $\sigma\colon R\to[n]$1. In the semidistributive case, $\sigma\colon R\to[n]$2 and $\sigma\colon R\to[n]$3, yielding the equality.

The converse is equally sharp: every echelon-independent lattice is semidistributive. This rules out the stronger but false expectation that $\sigma\colon R\to[n]$4-independence should be automatic for lattices in general. Within lattice theory, semidistributivity is exactly the criterion for independence of the chosen linear extension.

3. Trim lattices and vertebral linear extensions

Trim lattices provide a second major class in which echelonmotion agrees with rowmotion, but the mechanism is more selective. A trim lattice is an extremal lattice, meaning $\sigma\colon R\to[n]$5 and there is a maximal chain of length $\sigma\colon R\to[n]$6, that is also left-modular. Thomas–Williams defined rowmotion on any trim lattice; when the lattice is also semidistributive, this agrees with Barnard’s rowmotion. The result of Defant–Jiang–Marczinzik–Segovia–Speyer–Thomas–Williams is that every trim lattice admits a distinguished family of linear extensions for which echelonmotion coincides with rowmotion (Defant et al., 24 Jul 2025).

More precisely, one chooses a maximal chain

$\sigma\colon R\to[n]$7

and labels each cover $\sigma\colon R\to[n]$8 by the unique join-irreducible $\sigma\colon R\to[n]$9 such that $\sigma$0 and $\sigma$1 is minimal. For each $\sigma$2, one forms the word

$\sigma$3

and defines the vertebral linear extension $\sigma$4 by listing elements of $\sigma$5 in order of $\sigma$6 under lex order. Lemma 6.5 states that $\sigma$7 is indeed a linear extension.

Theorem 1.3 then asserts that if $\sigma$8 is trim and $\sigma$9 is any vertebral linear extension, one has

$\sigma(x)\le \sigma(y)$0

The proof again uses the pop-stack description of rowmotion and the fact that $\sigma(x)\le \sigma(y)$1 lies in the set of maximal elements of $\sigma(x)\le \sigma(y)$2. The decisive point is that, in the vertebral order, the unique maximal element of that set is exactly $\sigma(x)\le \sigma(y)$3. The induction proceeds through the combinatorics of the Galois graph of $\sigma(x)\le \sigma(y)$4 and the behavior of the chosen chain under intervals.

A useful distinction follows. On semidistributive lattices, echelonmotion agrees with rowmotion for every linear extension. On trim lattices, the theorem guarantees agreement only for vertebral linear extensions. The trim result is therefore not a general $\sigma(x)\le \sigma(y)$5-independence statement.

4. Involutivity on Eulerian posets

Echelonmotion has a particularly rigid form on Eulerian posets. A graded poset $\sigma(x)\le \sigma(y)$6 with rank function $\sigma(x)\le \sigma(y)$7 is Eulerian if

$\sigma(x)\le \sigma(y)$8

on every interval. For such posets, Defant–Jiang–Marczinzik–Segovia–Speyer–Thomas–Williams proved that for every linear extension $\sigma(x)\le \sigma(y)$9,

$x\le y$0

Thus echelonmotion is always an involution in the Eulerian case (Defant et al., 24 Jul 2025).

The matrix argument is direct. If $x\le y$1 and $x\le y$2 is the diagonal sign matrix with

$x\le y$3

then one checks that

$x\le y$4

If $x\le y$5 is its Bruhat factorization, then

$x\le y$6

is also a Bruhat factorization, so the permutation matrix used is still $x\le y$7. On the other hand,

$x\le y$8

shows that the Bruhat factorization of $x\le y$9 uses $R$0. Uniqueness therefore forces $R$1.

This involutivity statement is independent of any comparison with rowmotion. It applies to Eulerian posets as such, not only to lattices, and it holds for every linear extension. A plausible implication is that the Bruhat-theoretic definition retains substantial information from the Möbius-theoretic symmetry of Eulerian intervals even when no lattice operations are available.

5. Echelon-independence and MacNeille completion

A poset $R$2 is echelon-independent if for every pair of linear extensions $R$3 one has $R$4. In that case the poset carries a canonical bijection $R$5. For lattices, the classification is exact: a lattice is echelon-independent if and only if it is semidistributive. This equivalence is one of the central structural conclusions of the theory (Defant et al., 24 Jul 2025).

The paper also extends the analysis beyond lattices. Every echelon-independent connected poset is bounded. The proof idea given in the summary is that if $R$6 has two distinct maxima or two distinct minima, then one can find two linear extensions forcing echelonmotion to send the same element to two different extrema, contradicting independence. Moreover, if $R$7 is a connected echelon-independent poset, then its MacNeille completion $R$8 is a semidistributive lattice. The proof proceeds contrapositively: if $R$9 were not semidistributive, one produces two linear extensions of $\mathrm{Ech}_\sigma$00 inducing different echelonmotions. Consequently, any connected echelon-independent poset embeds into a semidistributive lattice $\mathrm{Ech}_\sigma$01 and inherits a natural rowmotion $\mathrm{Ech}_\sigma$02 echelonmotion.

The later paper "Short Proofs in Algebraic and Enumerative Combinatorics" adds a further sufficient condition: if the incidence algebra of $\mathrm{Ech}_\sigma$03 is Auslander-regular, then $\mathrm{Ech}_\sigma$04 is also echelon-independent. This does not replace the lattice-theoretic characterization, but it enlarges the class of posets known to exhibit $\mathrm{Ech}_\sigma$05-independence (Defant, 19 May 2026).

6. Modular lattices, cover statistics, and Dilworth’s theorem

A separate development concerns modular lattices. A finite lattice $\mathrm{Ech}_\sigma$06 is modular if for all $\mathrm{Ech}_\sigma$07 and all $\mathrm{Ech}_\sigma$08 one has

$\mathrm{Ech}_\sigma$09

Equivalently, in any modular lattice one has

$\mathrm{Ech}_\sigma$10

Dilworth’s classical result states that in a finite modular lattice the number of elements covering exactly $\mathrm{Ech}_\sigma$11 others equals the number of elements covered by exactly $\mathrm{Ech}_\sigma$12 others, for each $\mathrm{Ech}_\sigma$13. The conjecture of Defant–Jiang–Marczinzik–Segovia–Speyer–Thomas–Williams was that for every choice of $\mathrm{Ech}_\sigma$14, echelonmotion itself witnesses this equality degree by degree. That conjecture was resolved in "Short Proofs in Algebraic and Enumerative Combinatorics" (Defant, 19 May 2026).

For $\mathrm{Ech}_\sigma$15, write

$\mathrm{Ech}_\sigma$16

The theorem proved in the 2026 paper states that if $\mathrm{Ech}_\sigma$17 is a finite modular lattice and $\mathrm{Ech}_\sigma$18 is any linear extension, then

$\mathrm{Ech}_\sigma$19

The proof defines two upper-triangular matrices $\mathrm{Ech}_\sigma$20 and $\mathrm{Ech}_\sigma$21 by

$\mathrm{Ech}_\sigma$22

Using modularity, one checks that

$\mathrm{Ech}_\sigma$23

If $\mathrm{Ech}_\sigma$24 is the Bruhat factorization with $\mathrm{Ech}_\sigma$25, then

$\mathrm{Ech}_\sigma$26

and since conjugation by an upper-triangular matrix preserves diagonal entries, the permutation $\mathrm{Ech}_\sigma$27 transfers upward cover numbers to downward cover numbers.

Because $\mathrm{Ech}_\sigma$28 is a bijection, this immediately yields a bijection

$\mathrm{Ech}_\sigma$29

for each $\mathrm{Ech}_\sigma$30, recovering Dilworth’s theorem by cardinality. The paper emphasizes that this is a new algebraic bijective proof.

The examples highlighted in the summary situate the theorem within familiar classes. In the Boolean lattice $\mathrm{Ech}_\sigma$31, which is distributive, echelonmotion is the usual rowmotion, and on small $\mathrm{Ech}_\sigma$32 one checks the cover-size matching directly. The five-element nondistributive modular lattice $\mathrm{Ech}_\sigma$33 is the simplest nontrivial test case. More generally, the lattice of subspaces of a finite vector space over any field is modular but not distributive, so the theorem produces, for each linear extension, a degree-preserving permutation of the set of subspaces. A common point of confusion is to conflate this modular-lattice theorem with echelon-independence: the theorem asserts cover-count matching for every $\mathrm{Ech}_\sigma$34, whereas $\mathrm{Ech}_\sigma$35-independence itself remains characterized by semidistributivity.