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The Affine Tamari Lattice

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Pith's one-line read New Tamari lattices in affine type have Catalan-counted elements

desk verdict A substantial new family of finite lattices with a real but patchable proof gap in the semidistributivity of the affine member. read the letter →

arxiv 2502.07198 v1 pith:FX53P2AD submitted 2025-02-11 math.CO math.RAmath.RT

Grant Barkley , Colin Defant This is my paper · ORCID
classification math.COmath.RAmath.RT MSC 05A0506B0516G2020F55
keywords TamarilatticeaffinesymmetricgroupDyerordertranslation-invarianttotalordersnoncrossingarcdiagramstorsionclassesmaximalgreensequencescyclicsievingphenomenon
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

This paper establishes that the classical Tamari lattice has two finite affine analogues, obtained not from the usual weak order on affine permutations but from the Dyer order on translation-invariant total orders of the integers. Restricting to orders that avoid the pattern 312 yields the cyclic Tamari lattice $\mathrm{CTam}$ and, among real orders, the affine Tamari lattice $\mathrm{ATam}$. If correct, $|\mathrm{CTam}|$ equals the type-$B$ Catalan number $\binom{2n}{n}$ and $|\mathrm{ATam}|$ equals the type-$D$ Catalan number $\frac{3n-2}{n}\binom{2n-2}{n-1}$, and both lattices inherit quotient and sublattice structure from the infinite cyclic Dyer lattice. The same machinery proves that maximal green sequences for the completed path algebra of the oriented $n$-cycle have exactly the lengths in $[2n-1,\binom{n+1}{2}]$, and it describes the orbit structure of rowmotion through the cyclic sieving phenomenon. A sympathetic reader should care because this gives finite, Catalan-counted lattice models for affine Dyer order, with direct ties to cluster algebras, triangulations of punctured polygons, and the representation theory of Nakayama algebras.

What carries the argument

The central objects are translation-invariant total orders (TITOs): total orders $\preceq$ on $\mathbb{Z}$ with $a \preceq b$ iff $a+n \preceq b+n$. Their inversion sets $\mathrm{Inv}(\preceq)$, taken up to simultaneous translation, are ordered by inclusion to form the cyclic Dyer lattice $\mathrm{CDyer}$; the real TITOs, in which every singleton block is waxing, form the Dyer lattice. The argument is carried by the restriction to 312-avoiding TITOs, which are compact and therefore governed by their noncrossing arc diagrams of lower walls; the companion theorem that each compact TITO is uniquely determined by its diagram yields the Catalan counts. Alongside this, translation-invariant binary in-ordered trees (TIBITs) provide the mechanism for the lattice structure: the map sending a TITO to its binary insertion tree defines the cyclic sylvester congruence, under which $\mathrm{CTam}$ is the quotient, while the spine-flip maps realize $\mathrm{ATam}$ as a quotient of $\mathrm{CTam}$.

What would settle it

Enumerate the real 312-avoiding translation-invariant total orders for $n=4$; the paper predicts exactly $\frac{3n-2}{n}\binom{2n-2}{n-1}=50$ elements, so any other count—or a pair of distinct such orders with the same noncrossing lower-wall diagram—would refute the central bijection behind the lattice sizes.

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Extended reading notes

Core claim

The paper's central claim is that the cyclic Dyer lattice—the poset of all translation-invariant total orders on $\mathbb{Z}$ ordered by containment of inversion sets—contains a finite 312-avoiding sublattice $\mathrm{CTam}$, and that its real part contains a further quotient $\mathrm{ATam}$; both are genuine lattices with rich structure. The construction deliberately uses the Dyer order rather than the affine weak order, because the 312-avoiding affine permutations do not form a lattice. The authors prove that $\mathrm{CTam}$ is both a sublattice and a quotient of the cyclic Dyer lattice, that $\mathrm{ATam}$ is a quotient of $\mathrm{CTam}$ and of the Dyer lattice, and that both lattices are self-dual and semidistributive. They establish the cardinalities $\mathrm{Cat}_{B_n}$ and $\mathrm{Cat}_{D_n}$ through a bijection between 312-avoiding TITOs and noncrossing arc diagrams in an annulus, and they identify the Hasse diagram of $\mathrm{ATam}$ with the type-$D_n$ cluster exchange graph. They further show that rowmotion on these lattices has orbit structure governed by Kreweras complement and cyclic sieving, and that maximal green sequences for the associated algebras have exactly the predicted length intervals.

Load-bearing premise

The whole construction rests on the companion theorem, quoted rather than proved, that a compact translation-invariant total order is uniquely determined by the noncrossing picture of its downward cover relations; if that uniqueness failed, the Catalan counts and the rowmotion identification would lose their foundation.

Editorial extensions

If this is right

  • $\mathrm{CTam}$ and $\mathrm{ATam}$ are finite lattice quotients of the infinite cyclic Dyer lattice, so the otherwise-infinite affine Dyer order has finite 312-avoiding cores with Catalan cardinalities.
  • The Hasse diagram of $\mathrm{ATam}$ is isomorphic to the type-$D_n$ cluster exchange graph via triangulations of a punctured $n$-gon; consequently, cover relations in $\mathrm{ATam}$ are cluster mutations.
  • The lengths of maximal green sequences for the completed path algebra of the oriented $n$-cycle are exactly $[2n-1,\binom{n+1}{2}]$, and for the affine Tamari algebra exactly $[2n-2,\binom{n+1}{2}-1]$.
  • Rowmotion on $\mathrm{CTam}$ is dynamically equivalent to Kreweras complement on type-$B$ noncrossing partitions, exhibiting cyclic sieving with $\mathrm{Cat}_{B_n}(q)$; on $\mathrm{ATam}$ the orbit structure is likewise governed by an explicit cyclic sieving polynomial.
  • Both lattices are self-dual, semidistributive, and have $n$-regular Hasse diagrams, and their canonical join-and-meet decompositions are described explicitly by relations on annular arcs.

Reading between the lines

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

  • The same 312-avoiding TITO recipe might produce finite lattice quotients for other affine Coxeter types, since the Dyer-order framework is not specific to type $\widetilde{A}_{n-1}$; the obstruction is finding the right analogue of the compact-TITO bijection.
  • The maximal-green-sequence interval for $\mathrm{CQ}$ likely transfers to every finite-dimensional quotient of the oriented-cycle path algebra whose brick set coincides with that of $\mathrm{CQ}$, which would give a broader class of Nakayama algebras with known sequence lengths.
  • Because $\mathrm{CTam}$ and $\mathrm{ATam}$ are examples of non-trim semidistributive lattices with well-behaved rowmotion, their ornamentation and arc models are natural testing grounds for pop-stack operators and shard intersection orders, questions the authors list as future work.
  • The dynamical equivalence between rowmotion and Kreweras complement suggests that rowmotion on other semidistributive quotients of affine Dyer lattices could be analyzed through noncrossing partition dynamics, potentially yielding further cyclic sieving instances.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces two new finite lattices, the cyclic Tamari lattice CTam and the affine Tamari lattice ATam, defined as subposets of the cyclic Dyer lattice and the Dyer lattice of translation-invariant total orders (TITOs). It establishes that their cardinalities are the type B_n and type D_n Catalan numbers, that they are self-dual, and that they admit numerous combinatorial models: TIBITs, noncrossing arc diagrams, torsion classes of two Nakayama algebras, and type D triangulations. It also describes their FTFSDL factorizations, proves that the lengths of maximal chains are exactly the stated intervals, and analyzes rowmotion via the cyclic sieving phenomenon.

Significance. The paper proposes a natural affine analogue of the Tamari lattice, with rich incarnations and strong structural results. The main strengths are the explicit bijections between combinatorial models, the use of the Dyer lattice as a unifying framework, and the derived cyclic sieving results. If the central claims are correct, these lattices constitute new and interesting examples in the study of semidistributive lattices and their dynamics. The paper is generally well organized and the combinatorial constructions are presented with care, including several detailed examples (e.g., Example 3.2, Figure 6, Example 7.17). However, one load-bearing point in the proof of semidistributivity of ATam is not justified, and a key bijection is imported from an unpublished companion preprint, so the present version is not fully self-contained.

major comments (3)
  1. [Section 1; Theorem 4.15] The only stated justification for the semidistributivity of ATam is the sentence 'Because the cyclic Dyer lattice is semidistributive, we deduce that the cyclic and affine Tamari lattices are as well.' For CTam this is valid, since Theorem 4.13 proves that CTam is a sublattice of CDyer and semidistributivity is inherited by sublattices. However, for ATam the paper provides only Theorem 4.15, which makes ATam a lattice quotient of CTam. Homomorphic images of semidistributive lattices need not be semidistributive; for example, the diamond M3 fails semidistributivity but is a quotient of a free lattice. The paper offers no special property of the spine congruence to rule this out. This gap is load-bearing: Section 5 defines the map κ and the FTFSDL factorization of ATam before Theorem 5.5, and Section 9 defines rowmotion on ATam via Barnard's theorem, both of which require ATam to be semidistributive.
  2. [Section 6, Proposition 6.3] The proof of Proposition 6.3 does not establish the isomorphism between ATamTors and ATam312 independently. The proof provided for CTamTors relies on Theorem 5.5, whose statement for ATam already presupposes the unproved semidistributivity of ATam. The paper does not state or prove the affine analogue of Proposition 6.4 (the bijection between affine arc torsion classes and torsion classes for AQ), which is the natural way to obtain an independent proof. Therefore, Proposition 6.3 cannot serve as an alternative justification of semidistributivity. The authors should either prove directly that the spine congruence quotient preserves semidistributivity, or prove the bijection between ATam and ATamTors using the arc torsion class description from Remark 5.10 together with an affine version of Proposition 6.4, and then deduce semidistributivity from Lemma 6.2.
  3. [Section 5, Proposition 5.1] The bijection between 312-avoiding TITOs and noncrossing arc diagrams, which underlies the cardinality computation in Corollary 5.2 and the transfer of Kreweras-complement dynamics to rowmotion, is quoted from the companion preprint [6, Theorem 4.12] and is not proved in the present paper. The paper explicitly defers to [6] in the proof of Proposition 5.1. Since the companion is unpublished (though available on arXiv), the reader cannot verify this foundational step from the present manuscript alone. The authors should state the exact theorem from [6] that is being used and ideally summarize its proof, or at minimum clarify the status of [6] relative to this paper.
minor comments (4)
  1. [Section 3.2] In the paragraph defining pattern avoidance, the sentence 'We say ⪯ is 312-avoiding (respectively, 312-avoiding) if no three integers form a 312-pattern (respectively, a 312-pattern) in ⪯' appears garbled; the two displayed terms are likely meant to be different (e.g., '312-avoiding' and '132-avoiding'), and the intended distinction should be clarified.
  2. [Section 3.2, Lemma 3.7] The phrase 'any small enough a ∈ I2' is vague; it would be clearer to state that one selects a ∈ I2 with c−n < a < c, which exists because blocks have no minimal or maximal elements.
  3. [Section 9.4] The formula Row_{π↓≡(L)} = π↓≡ ∘ Row_L is stated without proof or reference. Since this formula is used to compute rowmotion on ATam from that on CTam, a proof or a precise citation to Barnard's work would help the reader.
  4. [Section 9.3, Proposition 9.2] The proof uses the fact that the Hasse diagram of CTam is n-regular (Proposition 4.14) to conclude |D(⪯)| + |D(Row(⪯))| = n. The argument is correct once one recalls that D(Row(⪯)) = U(Row(⪯)) by the definition of rowmotion, but this equality is not repeated here; a one-line clarification would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the paper's new lattice constructions and rowmotion results are proved from definitions, with foundational TITO facts imported from the authors' companion papers as independent support.

full rationale

The central claims—that the 312-avoiding TITOs form the cyclic and affine Tamari lattices, that these are sublattices and quotients of the cyclic Dyer lattice, that the TIBIT and arc-diagram models are isomorphic, and that rowmotion satisfies the stated cyclic sieving results—are derived in the text from explicit definitions and internal lemmas. The main external inputs are results from the authors' own prior work: Proposition 3.4 ([6]) gives semidistributivity of CDyer and Dyer, Proposition 3.5 ([6,7]) gives the cover-flip characterization, and Proposition 5.1 imports [6, Theorem 4.12] to assert that a compact TITO is uniquely determined by its noncrossing arc diagram. These are parameter-free theorems whose assumptions do not include the present paper's target conclusions, so under the stated rules they count as genuine evidence rather than circular reductions. One non-circular rigor gap should be noted: Section 1 states 'Because the cyclic Dyer lattice is semidistributive, we deduce that the cyclic and affine Tamari lattices are as well'; the affine part does not follow, since ATam is only a lattice quotient of CTam (Theorem 4.15) and quotients of semidistributive lattices need not be semidistributive. This leaves the FTFSDL and rowmotion results for ATam less rigorously supported, but this is a missing-proof/correctness concern, not a circularity. No fitted parameter is relabeled as a prediction, and no known pattern is merely renamed.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No numeric free parameters appear: the paper is purely combinatorial and algebraic. The new objects it introduces, such as TIBITs, arc torsion classes, and the cyclic and affine Tamari algebras, are explicitly constructed and proven bijective to existing or independently defined structures. The unproved inputs are the eight cited results listed above, several of which come from the authors' own companion papers.

assumptions (8)
  • standard math An equivalence relation on a complete lattice is a complete congruence iff its classes are intervals and the two projection maps are order-preserving (Proposition 2.1, from [54]).
    Used to conclude CTam and ATam are lattices by checking order preservation of the sylvester and spine projections in Theorems 4.12 and 4.15.
  • domain assumption CDyer and Dyer are completely semidistributive lattices, and the map pi_down_Dyer is a surjective complete lattice homomorphism (Proposition 3.4, quoted from [6]).
    This is the lattice-theoretic foundation for the quotient and sublattice claims; it is not proved in the present paper.
  • domain assumption Every cover in the Dyer or cyclic Dyer lattice is realized by a unique wall flip (Proposition 3.5, from [6,7]).
    Used throughout Sections 4 and 5 to relate cover relations to TIBIT edges and lower/upper walls.
  • domain assumption Every compact TITO is uniquely determined by its noncrossing arc diagram ([6, Theorem 4.12]).
    Essential for Proposition 5.1, which gives the bijection from 312-avoiding TITOs to noncrossing arc diagrams and hence the Catalan cardinalities.
  • standard math The Fundamental Theorem of Finite Semidistributive Lattices: a finite semidistributive lattice is determined by its join-irreducibles and the relations ->, ->, -> ([57]).
    Used in Sections 5 and 6 to describe CTam and ATam via canonical join representations and to compare with torsion-class lattices.
  • standard math For a finite-dimensional algebra of finite representation type, the torsion classes form a semidistributive lattice whose join-irreducibles are the bricks (Lemma 6.2, from [57]).
    Foundational for identifying CTam and ATam with torsion-class lattices of the cyclic and affine Tamari algebras.
  • domain assumption The algebras CQ and AQ are Nakayama algebras with finite representation type, and their indecomposable modules are exactly the string modules Ma,b ([1,11]).
    Used in Section 6 to translate between arcs, bricks, and torsion classes for the two algebras.
  • domain assumption The Armstrong-Stump-Thomas theorem: the type B noncrossing partitions under Kreweras complement exhibit cyclic sieving with CatBn(q) ([5]).
    Used in Theorem 9.3 to describe rowmotion on CTam and again in Section 9.4 to derive the ATam rowmotion orbit structure.

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Pith. "Pith review of The Affine Tamari Lattice." pith.science (2026). https://pith.science/paper/FX53P2AD

@misc{pith2026250207198,
  author       = {Pith},
  title        = {Pith review of: The Affine Tamari Lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FX53P2AD}},
  note         = {Machine review of arXiv:2502.07198}
}
abstract

Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which is the infinite lattice of translation-invariant total orders under containment of inversion sets. The affine Tamari lattice is a quotient of the Dyer lattice, which in turn is a quotient of the cyclic Dyer lattice and is isomorphic to the collection of biclosed sets of the root system of type $\widetilde{A}_{n-1}$ under inclusion. We provide numerous combinatorial and algebraic descriptions of these lattices using translation-invariant total orders, translation-invariant binary in-ordered trees, noncrossing arc diagrams, torsion classes, triangulations, and translation-invariant noncrossing partitions. The cardinalities of the cyclic and affine Tamari lattices are the Catalan numbers of types $B_n$ and $D_n$, respectively. We show that these lattices are self-dual and semidistributive, and we describe their decompositions coming from the Fundamental Theorem of Finite Semidistributive Lattices. We also show that the rowmotion operators on these lattices have well-behaved orbit structures, which we describe via the cyclic sieving phenomenon. Our new combinatorial framework allows us to prove that the lengths of maximal green sequences for the completed path algebra of the oriented $n$-cycle are precisely the integers in the interval $[2n-1,\binom{n+1}{2}]$.

Figures

Figures reproduced from arXiv: 2502.07198 by the authors.

Figure 1
Figure 1. The Hasse diagram of the affine Tamari lattice for n = 3, with each element represented as a window notation of a 312-avoiding translation-invariant total order. The darker elements are 312-avoiding affine permutations; the subposet that they induce is not a lattice. In Section 4, we introduce translation-invariant binary in-ordered trees (TIBITs) and a natural partial order on them that we also call the Dyer order.… view at source ↗
Figure 2
Figure 2. A translation-invariant binary in-ordered tree for n = 7 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Two different translation-invariant binary in-ordered trees for n = 4 whose underlying infinite binary trees are isomorphic. The next lemma is immediate from the definitions [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The cyclic Tamari lattice for n = 3. Each element is represented by a box containing a TIBIT T (on top), the corresponding 312-avoiding TITO ⊑T (on bottom), and the corresponding 132-avoiding TITO ⊑T (in the middle) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Applying π ↓ spine and π ↑ spine to two TIBITs for n = 5. For any TIBIT T, a reflection index (a, b) is an inversion of π ↓ spine(T) if and only if (a, b) is an inversion of T and a ̸≡ b (mod n). Similarly, (a, b) is a version of π ↑ spine(T) if and only if (a, b) is a…
Figure 6
Figure 6. Figure 6: The affine Tamari lattice for n = 3. Each element is represented by a box containing one or two TIBITs (on top), a 312-avoiding real TITO (on the bottom), and a 132-avoiding co-real TITO (in the middle) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: The arc diagrams of the 312-avoiding TITOs with window notations [1, 4, 3, 6, 5][7, 2] (left) and [1, 6, 5, 7, 4, 3][2] (right). Corollary 5.2. We have |CTam| = CatBn and |ATam| = CatDn . Proof. Let D be the set of noncrossing arc diagrams with no imaginary arcs. It fo…
Figure 8
Figure 8. Figure 8: For n = 3, each image shows a cylinder containing the join-irreducible elements of CTam. The left image shows the non-loop arrows of the form ↠ and ,→ in the FTFSDL factorization, while the right image shows all non-loop arrows of the form →. Deleting the parts of the …
Figure 9
Figure 9. Figure 9: On the left is the affine arc torsion class generated by the noncrossing arc diagram on the left of [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Some of the tagged arcs in P• for n = 5. Each tagged arc (blue) corresponds to an annular arc, which is overlaid in faint red. Definition 7.2. Let T be a torsion class in ATamTors. The sτ -tilting module of T is the unique AQ￾representation M ∈ T such that every indec…
Figure 11
Figure 11. Figure 11: On the left is the affine arc torsion class from the left side of [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: A TIBIT for n = 7 with three shaded principal order ideals [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 13
Figure 13. Figure 13: The maximum-length chain of O constructed in the proof of Theo￾rem 8.4 for n = 4. 9. Rowmotion 9.1. Dynamics. Consider a cyclic group Cω = ⟨gω⟩ of order ω acting on a finite set X. For F(q) ∈ C[q], we say the triple (X, Cω, F(q)) exhibits the cyclic sieving phenomenon…
Figure 14
Figure 14. Figure 14: In red is the type A6 noncrossing partition {{1, 3}, {2}, {4, 5, 7}, {6}}, whose Kreweras complement is {{1, 2}, {3, 7}, {4}, {5, 6}} (pink). We will need to consider three different types of noncrossing partitions. First, there are type AN−1 noncrossing partitions, w…
Figure 15
Figure 15. Figure 15: Let n = 7. On the top is a type Bn noncrossing partition ρ (red) together with Krew(ρ) # (pink). On the bottom is the TINCP R(ρ) (red) together with Krew(R(ρ))# (pink). There is a bijection R: TINC(Z) → NC(Bn) that simply reduces numbers modulo 2n. More precisely, for…
Figure 16
Figure 16. Figure 16: The TINCPs ρD and ρD′, where D and D′ are the noncrossing arc diagrams on the left and right, respectively, of [PITH_FULL_IMAGE:figures/full_fig_p038_16.png]
Figure 17
Figure 17. Figure 17: Applying π ↓ NC to a TINCP with n = 5. Proposition 9.2 and (5) together yield the following result [PITH_FULL_IMAGE:figures/full_fig_p039_17.png]
Figure 18
Figure 18. Figure 18: Applying π ↓ NC ◦ Krew−1 to a partition in Ξ0, where n = 8. Proposition 9.11. There is an action of the cyclic group C2n(n−1) = ⟨g2n(n−1)⟩ on Ξ satisfying g2n(n−1) · ρ = π ↓ NC(Krew(ρ)) for all ρ ∈ Ξ. Moreover, the triple  Ξ, C2n(n−1), [n] q 2(n−1)CatAn−2 (q ϵ(n) ) …

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Works this paper leans on

63 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [6]

    G. T. Barkley. Extended weak order for the affine symmetric group. Preprint arXiv:2502.05875

    arXiv

  2. [1]

    T. Adachi. The classification of τ -tilting modules over Nakayama algebras. J. Algebra, 452 (2016), 227–262

    work page 2016

  3. [2]

    Adachi, O

    T. Adachi, O. Iyama and I. Reiten. τ -tilting theory. Compos. Math., 150 (2014), 415–452

    work page 2014

  4. [3]

    Ajran and C

    K. Ajran and C. Defant. The pop-stack operator on ornamentation lattices. Preprint: arXiv:2501.10311

    arXiv

  5. [4]

    Apruzzese and K

    P.J. Apruzzese and K. Igusa. Stability conditions for affine type A. Algebr. Represent. Theory, 23 (2020), 2079– 2111

    work page 2020

  6. [5]

    Armstrong, C

    D. Armstrong, C. Stump, and H. Thomas. A uniform bijection between nonnesting and noncrossing partitions. Trans. Amer. Math. Soc. , 365 (2013), 4121–4151

    work page 2013

  7. [7]

    G. T. Barkley and D. E. Speyer. Combinatorial descriptions of biclosed sets in affine type. Comb. Theory , 4 (2024)

    work page 2024

  8. [8]

    G. T. Barkley and D. E. Speyer. Affine extended weak order is a lattice. Preprint arXiv:2311.05737

    arXiv

Show all 63 references
  1. [9]

    E. Barnard. The canonical join complex. Electron. J. Combin. , 26 (2019)

    2019

  2. [10]

    Barnard, C

    E. Barnard, C. Defant, and E. J. Hanson. Pop-stack operators for torsion classes and Cambrian lattices. Preprint arXiv:2312.03959

    arXiv

  3. [11]

    Baur and R

    K. Baur and R. Coelho Sim˜ oes. A geometric model for the module category of a string algebra. Preprint arXiv:2403.07810

    arXiv

  4. [14]

    Bousquet-M´ elou, E

    M. Bousquet-M´ elou, E. Fusy, and L.-F. Pr´ eville-Ratelle. The number of intervals in the m-Tamari lattices. Electron. J. Combin. , 18 (2011)

    2011

  5. [15]

    Br¨ ustle, D

    T. Br¨ ustle, D. Smith, and H. Treffinger. Wall and chamber structure for finite-dimensional algebras.Adv. Math., 354 (2019)

    2019

  6. [16]

    Br¨ ustle and D

    T. Br¨ ustle and D. Yang. Ordered exchange graphs, in Advances in representation theory of algebras , 135–193, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨ urich

  7. [17]

    von Bell, R

    M. von Bell, R. S. Gonz´ alez D’Le´ on, F. A. Mayorga Cetina, and M. Yip. A unifying framework for theν-Tamari lattice and principal order ideals in Young’s lattice. Combinatorica, 43 (2023), 479–504

    2023

  8. [18]

    Bousquet-M´ elou, G

    M. Bousquet-M´ elou, G. Chapuy, and L.-F. Pr´ eville-Ratelle. The representation of the symmetric group on m- Tamari intervals. Adv. Math., 247 (2013), 309–342

    2013

  9. [19]

    Ceballos and C

    C. Ceballos and C. Chenevi` ere. On linear intervals in the alt ν-Tamari lattices. Comb. Theory, 4 (2024)

    2024

  10. [20]

    Ceballos, A

    C. Ceballos, A. Padrol, and C. Sarmiento. Geometry of ν-Tamari lattices in types A and B. Trans. Amer. Math. Soc., 371 (2019), 2575–2622

    2019

  11. [21]

    Ceballos and V

    C. Ceballos and V. Pilaud. The diameter of type D associahedra and the non-leaving-face property. European J. Combin. , 51 (2016), 109–124

    2016

  12. [22]

    Chapoton, S

    F. Chapoton, S. Fomin, and A. Zelevinsky. Polytopal realizations of generalized associahedra. Canad. Math. Bull., 45 (2002), 537–566

    2002

  13. [23]

    Chenevi` ere

    C. Chenevi` ere. Linear intervals in the Tamari and the Dyck lattices and in the alt-Tamari posets. Preprint arXiv:2209.00418

    arXiv

  14. [24]

    Choi and N

    Y. Choi and N. Sun. The image of the Pop operator on various lattices. Adv. Appl. Math. , 154 (2024)

    2024

  15. [25]

    A. Crites. Enumerating pattern avoidance for affine permutations. Electron. J. Combin. , 17 (2010)

    2010

  16. [26]

    C. Defant. Pop-stack-sorting for Coxeter groups. Comb. Theory, 2 (2022)

    2022

  17. [27]

    C. Defant. Meeting covered elements in ν-Tamari lattices. Adv. Appl. Math. , 134 (2022)

    2022

  18. [28]

    Defant and J

    C. Defant and J. Lin, Rowmotion on m-Tamari and biCambrian lattices. Comb. Theory, 4 (2024)

    2024

  19. [29]

    Defant and A

    C. Defant and A. Sack. Operahedron lattices. Preprint arXiv:2402.12717

    arXiv

  20. [30]

    Defant and N

    C. Defant and N. Williams. Semidistrim lattices. Forum Math. Sigma , 11 (2023)

    2023

  21. [31]

    Demonet, O

    L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas. Lattice theory of torsion classes: beyond τ -tilting theory. Trans. Amer. Math. Soc. Ser. B , 10 (2023), 542–612

    2023

  22. [32]

    M. Dyer. Quotients of twisted Bruhat orders. J. Algebra, 163 (1994), 861–879

    1994

  23. [33]

    M. Dyer. On the weak order of Coxeter groups. Canad. J. Math. , 71 (2019), 299–336

    2019

  24. [34]

    J. Fei. Tropical F -polynomials and general presentations. J. Lond. Math. Soc. , 107 (2023), 2079–2120

    2023

  25. [35]

    Fomin, L

    S. Fomin, L. Williams, and A. Zelevinsky. Introduction to cluster algebras. Chapters 4-5. Preprint arXiv:1707.07190

    arXiv

  26. [36]

    Freese, J

    R. Freese, J. Jeˇ zek, and J. B. Nation. Free lattices, volume 42 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995

    1995

  27. [37]

    Garver and T

    A. Garver and T. McConville. Lattice properties of oriented exchange graphs and torsion classes. Algebr. Rep- resent. Theory, 22 (2019), 43–78. 45

    2019

  28. [38]

    Garver, T

    A. Garver, T. McConville, and K. Serhiyenko. Minimal length maximal green sequences. Adv. in Appl. Math. , 96 (2018), 76–138

    2018

  29. [39]

    Gorsky and N

    M. Gorsky and N. J. Williams. A structural view of maximal green sequences. Preprint: arXiv:2301.08681

    arXiv

  30. [40]

    L. Hong. The pop-stack-sorting operator on Tamari lattices. Adv. Appl. Math. , 139 (2022)

    2022

  31. [41]

    S. Hopkins. The CDE property for skew vexillary permutations. J. Combin. Theory, Ser. A , 168 (2019)

    2019

  32. [42]

    S. Hopkins. Order polynomial product formulas and poset dynamics. In: Open Problems in Algebraic Combina- torics, Proceedings of Symposia in Pure Mathematics, 110, 2024

    2024

  33. [43]

    Iyama and R

    O. Iyama and R. Marczinzik. Distributive lattices and Auslander regular algebras. Adv. Math., 398 (2022)

    2022

  34. [44]

    Keller and L

    B. Keller and L. Demonet. A survey on maximal green sequences, in Representation theory and beyond, 267–286, Contemp. Math., 758, Amer. Math. Soc., RI

  35. [45]

    R. Laking. String algebras in representation theory, Ph.D. Thesis, University of Manchester, 2016

    2016

  36. [46]

    Marczinzik, H

    R. Marczinzik, H. Thomas, and E. Yıldırım. On the interaction of the Coxeter transformation and the rowmotion bijection. J. Comb. Algebra , 8 (2024), 359–374

    2024

  37. [47]

    H. M¨ uhle. The core label order of a congruence-uniform lattice. Algebra Universalis, 80 (2019)

    2019

  38. [48]

    Muller-Hoissen, J

    F. Muller-Hoissen, J. M. Pallo, and J. Stasheff, editors. Associahedra, Tamari Lattices and Related Structures. Tamari Memorial Festschrift, volume 299 of Progress in Mathematics. Springer, New York, 2012

    2012

  39. [49]

    Nakanishi and A

    T. Nakanishi and A. V. Zelevinsky, On tropical dualities in cluster algebras, in Algebraic groups and quantum groups, 217–226, Contemp. Math., 565, Amer. Math. Soc., Providence, RI

  40. [50]

    Padrol, V

    A. Padrol, V. Pilaud, and J. Ritter. Shard polytopes. Int. Math. Res. Not. IMRN , 2023 (2023), 7686–7796

    2023

  41. [51]

    Pilaud and Y

    V. Pilaud and Y. Palu. Non-kissing complexes and tau-tilting theory for gentle algebras. Mem. Amer. Math. Soc., 274 (2021)

    2021

  42. [52]

    Pr´ eville-Ratelle and X

    L.-F. Pr´ eville-Ratelle and X. Viennot. An extension of Tamari lattices. Trans. Amer. Math. Soc. 369 (2017), 5219–5239

    2017

  43. [53]

    N. Reading. Cambrian lattices. Adv. Math., 205 (2006), 313–353

    2006

  44. [54]

    N. Reading. Lattice theory of the poset of regions, in Lattice Theory: Special Topics and Applications, vol. 2 . Birkh¨ auser, 2016

    2016

  45. [55]

    N. Reading. Noncrossing arc diagrams and canonical join representations. SIAM J. Discrete Math. , 29 (2015)

    2015

  46. [56]

    N. Reading. Noncrossing partitions and the shard intersection order. J. Algebraic Combin. , 33 (2011), 483–530

    2011

  47. [57]

    Reading, D

    N. Reading, D. E. Speyer, and H. Thomas. The fundamental theorem of finite semidistributive lattices. Selecta Math., 27 (2021)

    2021

  48. [58]

    Rognerud

    B. Rognerud. The bounded derived categories of the Tamari lattices are fractionally Calabi–Yau. Adv. Math. , 389 (2021)

    2021

  49. [59]

    Schiffler

    R. Schiffler. A geometric model for cluster categories of type Dn. J. Algebraic. Combin. , 27 (2008), 1–21

    2008

  50. [60]

    J. Striker. Dynamical algebraic combinatorics: promotion, rowmotion, and resonance. Notices Amer. Math. Soc., 64 (2017), 543–549

    2017

  51. [61]

    Striker and N

    J. Striker and N. Williams. Promotion and rowmotion. European J. Combin., 33 (2012), 1919–1942

    2012

  52. [62]

    Stump, H

    C. Stump, H. Thomas, and N. Williams. Cataland: Why the Fuß? To appear in Mem. Amer. Math. Soc

  53. [63]

    D. Tamari. The algebra of bracketings and their enumeration. Nieuw Archief voor Wiskunde, 10 (1962), 131–146

    1962

  54. [64]

    H. Thomas. An analogue of distributivity for ungraded lattices. Order 23 (2006), 249–269

    2006

  55. [65]

    Thomas and N

    H. Thomas and N. Williams. Rowmotion in slow motion. Proc. Lond. Math. Soc. , 119 (2019), 1149–1178. Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Email address : gbarkley@math.harvard.edu Department of Mathematics, Harvard University, Cambridge, MA 0...

    2019

Pith tools

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