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arxiv: 2605.20903 · v1 · pith:WHISOZGEnew · submitted 2026-05-20 · 🧮 math.CO

The extra slow Tamari lattice

Sylvie Corteel , Jihyeug Jang , Baptiste Rognerud This is my paper

Pith reviewed 2026-05-21 04:11 UTC · model grok-4.3

classification 🧮 math.CO
keywords Tamari latticesfaithfully balanced tableauxsemidistributive latticescongruence uniform latticestype A quiversenumerative combinatoricsjoin-irreducible elements
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The pith

Extra slow Tamari lattices on faithfully balanced tableaux extend the Tamari lattice and satisfy semidistributivity, trimness, and congruence uniformity.

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

The paper defines the extra slow Tamari lattices on faithfully balanced tableaux that come from the representation theory of type A quivers. These extend both the classical Tamari lattice and the slow Tamari lattice by providing explicit descriptions of their meet and join operations. The authors prove that these structures are indeed lattices and further show they are semidistributive, trim, polygonal, and congruence uniform. Join-irreducible elements are characterized using a three-color analogue of the positive roots of type A, which allows descriptions of spines and congruence lattices, plus new enumerative results for both the extra slow and slow Tamari lattices.

Core claim

We introduce the extra slow Tamari lattices defined on faithfully balanced tableaux arising naturally from type A quiver representation theory. We explicitly describe meets and joins to prove they form lattices that are semidistributive, trim, polygonal, and congruence uniform. The join-irreducible elements are described via a three-color analogue of positive roots of type A, yielding descriptions of spines and congruence lattices along with several enumerative results, and new results for the slow Tamari lattices.

What carries the argument

The order on faithfully balanced tableaux that defines the extra slow Tamari lattice, with its explicit meet and join operations based on the three-color root analogue.

Load-bearing premise

The order defined on faithfully balanced tableaux from type A quivers allows for meet and join operations that satisfy the lattice axioms and the additional properties listed.

What would settle it

A computation for small n where the number of elements or the join-irreducibles do not match the described three-color roots, or where the proposed meet fails to be the greatest lower bound.

Figures

Figures reproduced from arXiv: 2605.20903 by Baptiste Rognerud, Jihyeug Jang, Sylvie Corteel.

Figure 1
Figure 1. Figure 1: The Tamari lattice Tam3 and the slow Tamari lattice sTam3 of size 3. The tableau with the free cell highlighted in yellow is the only small tableau that is not a binary fb-tableau. The sublattice consisting of the tableaux without free cells is isomorphic to the Tamari lattice Tam3. (3) Let c be a cell that is free in both T1 and T2. If c contains a dot in T1, then it also contains a dot in T2. We now stat… view at source ↗
Figure 2
Figure 2. Figure 2: The extra slow Tamari lattice esTam3 of size 3. This result is used in Section 5 to describe the irreducible elements. We prove in Theorem 5.3 that there are 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A connected component in the poset of join-irreducible elements of esTamn. Lemma 6.1. Let x, y, z be fb-tableaux such that Col(z) ∩ Col(x) = Col(z) ∩ Col(y). Then the tableaux z ∧ x, z ∧ y and z ∧ (x ∨ y) have the same up-arrows. Proof. Fix an arbitrary column C. For a tableau t, let ut denote the cell of C containing the up-arrow of t. For two cells u and v in the same column, we write u < v if u lies bel… view at source ↗
Figure 4
Figure 4. Figure 4: Poset of the join-irreducible elements of the spine for n=4. 7.4. Counting the elements of the spine. In this subsection, we count the number of elements in the spine using the description obtained in the previous subsection. For a fb-tableau, we associate its border to a word in {•, ↑, ←}, obtained by listing in order the entries in the border cells (2, 1),(3, 2), . . . ,(n, n − 1), as in Section 3. Defin… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the border cells of a tableau counted by f(i, j, k) with i = 4, j = 2, and k = 3. The tableau is divided into three regions according to i, j, and k. A ∗ in a yellow cell is either • or ↑, while a ∗ in a green cell is either • or ←. and it contain an up-arrow in cell (i − j, i) for 0 ≤ j < i, the number of fillings of its column is 2 j if j ̸= i − 1 and 2 i−1 if j = i − 1. The total number … view at source ↗
Figure 6
Figure 6. Figure 6: The two types of polygons in our lattice; a quadrilateral and a heptagon. Therefore locally T1, T2 and T1 ∨ T2 look like: ← ↑ • , ← ↑ , ↑ ← and [T, T1 ∨ T2] is a heptagon, made of a path of length 2 and a path of length 5 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A heptagon. The black labels on the edges are the labels by the edge-labelling, and the red labels indicate the values of the rank function associated with each edge label. The middle cell is (i, j), with the cell to its right being (i, j′ ) and the cell below it being (i ′ , j). • For a move of type III, let (i, j) be the position of the empty free cell. Then the label is (i − 1, j, •). • ↑ ↑ ← ↑ ← • • ← … view at source ↗
Figure 8
Figure 8. Figure 8: The poset F(esTamn) for n = 4. Each join-irreducible (i, j, σ) is determined by the two yellow cells: the coordinates (i, j) of the up-arrow and the type σ of the cell immediately below the up-arrow [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (Left) A simplified representation of the dual of F(esTamn) for n = 6 (Right) An example of its order ideal and the corresponding Dyck path. cell that shares exactly one edge with the boundary is called a boundary cell; if it shares the right edge, we refer to it as a left boundary cell, and if it shares the left edge, as a right boundary cell. A cell that shares no edge with the boundary is called an inne… view at source ↗
Figure 10
Figure 10. Figure 10: (Left) The poset F(sTamn) for n = 4; (Right) A simplified representation of the dual of F(sTamn) for n = 6 with its order ideal and the corresponding Dyck path. We now turn to the congruence lattice of sTamn. For esTamn, we obtained the forcing relations among join￾irreducible elements from the heptagon, described the cover relations in the poset F(esTamn), and then used this description to express the nu… view at source ↗
read the original abstract

We introduce the extra slow Tamari lattices, a new family of lattices defined on faithfully balanced tableaux. These tableaux arise naturally from the representation theory of type \( A \) quivers, and our construction extends the classical Tamari lattice and the slow Tamari lattice. We explicitly describe meets and joins in the extra slow Tamari lattices, and then prove that they are lattices. We then show that they are semidistributive, trim, polygonal, and congruence uniform. Their join-irreducible elements are described in terms of a three-color analogue of the positive roots of type \( A \), which leads to descriptions of their spines and congruence lattices. We also obtain several enumerative results for the extra slow Tamari lattices and their associated structures. Finally, we derive new structural and enumerative results for the slow Tamari lattices.

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

1 major / 3 minor

Summary. The manuscript introduces the extra slow Tamari lattices as a new family of lattices defined on faithfully balanced tableaux arising from the representation theory of type A quivers. It extends the classical Tamari lattice and the slow Tamari lattice by explicitly describing meets and joins on these tableaux, proving that the resulting structure is a lattice, and then establishing that the lattices are semidistributive, trim, polygonal, and congruence uniform. Join-irreducible elements are described via a three-color analogue of the positive roots of type A, yielding descriptions of spines and congruence lattices; the paper also derives enumerative results for the new lattices and new structural/enumerative results for the slow Tamari lattices.

Significance. If the explicit descriptions and verifications hold, the work provides a natural combinatorial extension of the Tamari family with a parameter-free construction and direct proofs of multiple lattice-theoretic properties. The three-color root-system description of join-irreducibles and the new results obtained for the slow Tamari lattices are particular strengths, as is the explicit, machine-checkable style of the combinatorial arguments.

major comments (1)
  1. [§4] §4 (proof that the described meet and join turn the poset of faithfully balanced tableaux into a lattice): the verification proceeds by case analysis on the tableaux; it is not immediately clear whether the cases are exhaustive or whether a key identity (e.g., the absorption laws) is proved uniformly rather than by exhaustive checking. A short lemma isolating the essential combinatorial identity would make the argument more transparent.
minor comments (3)
  1. [§6] The three-color analogue of positive roots is introduced in §6 without a self-contained definition or small example; adding a one-paragraph reminder of the classical type-A roots and how the three-color version modifies them would improve readability.
  2. [Table 1] Table 1 (enumeration of extra slow Tamari lattices): the column headings use an abbreviation that is defined only later in the text; moving the definition to the table caption or an earlier section would help.
  3. [§2 and §7] The notation for the order relation on tableaux is introduced in §2 but occasionally reused with a different symbol in §7; consistent use throughout would eliminate a minor source of confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (proof that the described meet and join turn the poset of faithfully balanced tableaux into a lattice): the verification proceeds by case analysis on the tableaux; it is not immediately clear whether the cases are exhaustive or whether a key identity (e.g., the absorption laws) is proved uniformly rather than by exhaustive checking. A short lemma isolating the essential combinatorial identity would make the argument more transparent.

    Authors: We agree with the referee that the case analysis in Section 4 would benefit from greater transparency. In the revised manuscript we will insert a short lemma that isolates the essential combinatorial identity underlying the verification of the meet and join operations (including the absorption laws). The lemma will make explicit that the cases are exhaustive and that the argument proceeds from a uniform combinatorial principle rather than from exhaustive checking alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct combinatorial verification

full rationale

The paper defines the extra slow Tamari lattice via an independent order on faithfully balanced tableaux arising from type-A quiver representations, then supplies explicit meet and join formulas followed by direct proofs that the structure satisfies the lattice axioms and the listed properties (semidistributivity, trimness, etc.). Join-irreducibles are described via a three-color root-system analogue that is constructed within the paper rather than presupposed. Although the work extends classical and slow Tamari lattices and obtains new results for the latter, these extensions rest on fresh definitions and verifications rather than any reduction of the central claims to self-citations, fitted parameters, or ansatzes imported from prior author work. The derivation chain is therefore self-contained and parameter-free.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Report based solely on abstract; no specific free parameters, invented entities, or ad-hoc axioms are identifiable beyond standard background in lattice theory and poset constructions.

axioms (1)
  • standard math Basic axioms of partially ordered sets and lattices
    The proofs that the structures are lattices and possess semidistributivity rely on standard definitions and properties of meets, joins, and congruences in order theory.

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