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Relative Interval Tilting, Higher Auslander Staircase Corners and Rational Dyck Posets

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Pith's one-line read For coprime positive integers, the incidence algebra of a rational Dyck poset tensored with a line quiver is derived equivalent to the full lattice of lattice paths, proving a conjecture of Chapoton, Ladkani, and Rognerud.

desk verdict A substantial paper with a genuinely new relative tilting theorem and a Dyck-corner equivalence that looks right; the CLR proof is conditional on Xing's unverified preprint. read the letter →

arxiv 2608.06696 v1 pith:C3M6GV4I submitted 2026-08-07 math.RT math.CT

Shengyong Pan This is my paper · ORCID
classification math.RTmath.CT MSC 16G2018G8005E1653D37
keywords derivedequivalenceintervaltiltingstaircaseposethigherAuslanderalgebrarationalDyckpathpartiallywrappedFukayacategoryChapoton–Ladkani–RognerudconjectureKanextension
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 proves the Chapoton–Ladkani–Rognerud conjecture for every pair of coprime positive integers $a,b$: the incidence algebra of the product of a type-$A$ line quiver with the rational Dyck poset $Dyck_{a,b}$ is derived equivalent to the incidence algebra of the full lattice $L_{a,b}$ of lattice paths in the $a\times b$ rectangle. The proof supplies the missing link in an existing chain: a tilt from the Dyck poset's incidence algebra to Xing's algebra $B_0$, obtained as the canonical corner of a higher Auslander algebra of type $A$. The main tool is a relative interval-tilting theorem that extends the Chapoton–Ladkani–Rognerud mechanism from incidence categories of posets to arbitrary finite $k$-linear categories, tolerating higher-dimensional Hom spaces, non-semisimple diagonal algebras, and zero compositions. A reader should care because the equivalence connects Catalan-indexed poset combinatorics with representation theory and symplectic geometry, and the method produces explicit tilting complexes rather than an existence proof.

What carries the argument

The engine is the relative interval-tilting theorem (Theorem 3.9). Given a finite $k$-linear category $X$, a finite poset $Y$, and a monotone family of full sieves $F(y)\subseteq X$, it builds two categories $\Gamma(X,Y,F)$ and $\Gamma^\sharp(X,Y,F)$, the second differing by a 'target lies in the source fibre' condition that forces certain composites to vanish. The tilting object is the sum over $y\in Y$, $x\in F(y)$ of exact right Kan extensions $(\iota_y)_* \mathrm{Hom}_{F(y)}(x,-)$, and the theorem computes its opposite endomorphism category as $\Gamma^\sharp$; Rickard's derived Morita theorem then gives the equivalence. Iterating this construction coordinate by coordinate inserts one interlacing inequality and its forced-zero composites at each step, and the last category is the corner of a higher Auslander algebra. A forced-prefix deletion identifies rational Dyck paths with such staircases, and the resulting corner is shown to be Xing's $B_0$.

What would settle it

Compute the corner algebra and Xing's $B_0$ for a coprime pair not covered by explicit examples, say $(a,b)=(5,6)$: if the quiver with relations of $B_0$ differs from the staircase category $C_{m-1}(\Omega_{a,b})$ in any forced-zero composite, Proposition 5.5 and Theorem 6.6 collapse. Alternatively, find a coprime pair where the Cartan matrices of the two sides fail to satisfy the derived-invariance checks used in Section 8.

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Extended reading notes

Core claim

The central claim is Theorem 6.6: for coprime positive integers $a,b$ and a field $k$, $D^b(k(\vec A_{a+b}\times Dyck_{a,b}^{above})) \simeq D^b(kL_{a,b})$. The discovery that carries the argument is that every finite coordinate staircase admits a derived equivalence to a higher Auslander corner: iterating the relative interval-tilting theorem one coordinate at a time yields $D^b(k\Omega(H)) \simeq D^b(e_\Omega A e_\Omega)$ for every finite staircase $\Omega(H)$. For rational Dyck staircases this corner is exactly Xing's $B_0$ after a forced-prefix deletion, and the above/below convention is tracked by taking an opposite algebra. The new Dyck-corner equivalence fills the gap that the previously known results left open, and only the final identification with $L_{a,b}$ requires coprimality.

Load-bearing premise

The proof depends on accepting two cited equivalences from Xing's work—that the corner algebra built from Dyck paths is exactly $B_0$, and that $B_0$'s replicated version is derived equivalent to the relevant higher Auslander algebra—neither of which is reproved here.

Editorial extensions

If this is right

  • The Chapoton–Ladkani–Rognerud conjecture is true for all coprime $a,b$, so the incidence algebra of every rational Dyck poset, after tensoring with a line quiver, is derived equivalent to the full lattice algebra.
  • Every finite coordinate staircase incidence algebra is derived equivalent to an idempotent corner of a higher Auslander algebra of type $A$, with an explicitly constructed tilting complex.
  • The Dyck-corner equivalence requires no coprimality and is compatible with replicated algebras: $D^b(k\vec A_r \otimes k Dyck_{a,b}) \simeq D^b((B^{st}_{a,b})^{(r)})$ and its opposite version for the above-diagonal convention.
  • Staircase derived categories embed as thick subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products; for coprime Dyck parameters the same category is modeled by Fukaya–Seidel categories of symmetric Brieskorn–Pham singularities.
  • Along the equivalence the Serre functor of the full path lattice is transported to rotation of the stopped disk, yielding fractional Calabi–Yau dimension $ab/(a+b+1)$.

Reading between the lines

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

  • Going beyond the paper, the relative tilting theorem likely applies to sieve-indexed families in categories that are not directed Schur, so it may produce tilting equivalences for many incidence-like algebras with zero relations outside the staircase family.
  • Going beyond the paper, replicating the Dyck-corner equivalence in non-coprime cases suggests that an orbit-weighted presilting object might extend the CLR equivalence to all $a,b$; the paper itself notes generation remains open there.
  • Going beyond the paper, the Fukaya realization makes a concrete prediction: if the twisted complexes $L^{(j)}_v$ can be realized by embedded Lagrangians, the half-square zero relations should correspond to holomorphic polygons meeting the stop, a checkable symplectic computation.
  • Going beyond the paper, the explicit Cartan and Coxeter checks for the $(3,4)$ and $(4,5)$ staircases suggest that computing Coxeter polynomials of $C_j(\Omega_{a,b})$ at every tilting step for larger pairs would test whether the local corners inherit fractional Calabi–Yau periodicity.
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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 / 5 minor

Summary. The paper develops a relative interval-tilting theorem for finite k-linear categories with arbitrary finite-dimensional Hom spaces, non-semisimple diagonal endomorphism algebras, and forced-zero composites. It iterates this theorem to establish derived equivalences between incidence algebras of finite coordinate staircases and idempotent corners of higher Auslander algebras of type A. It then identifies rational Dyck posets with such staircases, proves that the resulting corner is Xing's algebra B0 in the coprime case, and combines the new Dyck-corner equivalence with results of Ladkani, Xing, and Gottesman to prove the Chapoton--Ladkani--Rognerud conjecture for coprime positive integers. The paper also gives Fukaya-categorical interpretations of the staircase equivalences and includes explicit finite checks for the (3,4) and (4,5) examples.

Significance. If the external inputs are correct, Theorem 6.6 resolves the CLR conjecture. The paper's own contribution is substantial: the relative tilting theorem genuinely extends the CLR interval-tilting mechanism to categories with higher-dimensional Hom spaces and zero composites, the iterated staircase theorem is explicit and does not require coprimality, and the non-coprime replicated statements are new. The paper is also commendably explicit about the division between new results and cited ones: Section 8 provides reproducible finite checks (Cartan determinants, Coxeter polynomials, and complete composable-pair counts) that are consistent with the staircase and Dyck-corner constructions. The main weakness is that the final CLR proof depends at two load-bearing points on results from the recent preprint [14] that are cited but not proved, and on a one-paragraph self-oppositeness lemma; these need to be addressed before the central claim can be regarded as fully established.

major comments (3)
  1. [Section 6.d, proof of Theorem 6.6] The proof of Theorem 6.6 is a chain whose final junction passes through two results from the preprint [14] that are not proved or even stated in this paper: the identification B0 ≅ eA'e in Proposition 5.5 (citing [14, Proposition 4.33]) and the derived equivalence B0^{(a+b)} ≃ A^a_{b+1} used in the proof of Theorem 6.6 (citing [14, Theorem 4.5 and Proposition 4.25]). The new Dyck-corner equivalence of Corollary 5.8 alone does not reach kL_{a,b}; without the Xing junction the chain stops at B0^op. Because Theorem 6.6 is the central claim, this dependence is load-bearing. The finite checks in Section 8 validate the staircase and Dyck-corner constructions but do not independently verify the Xing junction. Please either include proofs or detailed convention-matching derivations of these two external results, or explicitly state that the main theorem is conditional on [14] and give the precise statements and version used.
  2. [Lemma 6.5] The proof of self-oppositeness of A^a_{b+1} is a one-paragraph sketch. It asserts that σ(z) = (a+b+1-z_a, ..., a+b+1-z_1) sends arrows to arrows in the opposite direction and preserves the zero relations, but it does not verify the strict interlacing inequalities under σ, nor does it check the half-square zero-product rule for two composable arrows whose outer pair fails to interlace. Since this lemma is used in the proof of Theorem 6.6 to identify (A^a_{b+1})^op with A^a_{b+1}, it is load-bearing. Please provide a complete proof, for example by explicitly verifying that the interlacing inequalities are reversed appropriately under σ and by checking the composite rule (8) under the reversal.
  3. [Proposition 5.5 / Corollary 5.8] The identification of the staircase corner with Xing's B0 depends on matching the vertex labeling of rational Dyck paths with the indexing used in [14, Proposition 4.33]. The paper correctly tracks the above/below convention and takes opposites in Corollary 5.8, but because this identification is cited rather than proved, a mismatch in coordinate order or in the direction of the path order would propagate through the proof of Theorem 6.6 and invalidate the conclusion. Please include an explicit translation between the coordinates β(x) used here and Xing's vertex set for B0, or state precisely which statement in [14] is being used and why it applies verbatim to the conventions of this paper.
minor comments (5)
  1. [Section 7.e] The phrase 'By Theorem 7.4' should read 'By Corollary 7.4'.
  2. [Acknowledgments] The acknowledgments contain a nonstandard poetic passage beginning 'For many years, the fundamental axiom of my mathematical creed...'; this is inappropriate for a research paper and should be removed.
  3. [Definition 4.1] The sentence 'The second family of inequalities is empty when j = 0' is unclear; it should say that the interlacing family of inequalities is empty when j = 0.
  4. [Equation (85)] The symbol C_j is used both for the category in Definition 4.1 and for the Cartan matrix in Eq. (85); please disambiguate, for example by writing Cart_j for the Cartan matrix.
  5. [Abstract and Introduction] The abstract and introduction state that the paper proves the Chapoton--Ladkani--Rognerud conjecture; in light of the dependence on [14] noted above, this claim should be qualified or the external results should be proved.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new Dyck-corner equivalence is proved from first principles; the final chain merely composes cited external equivalences.

full rationale

The derivation is self-contained through the new staircase/Dyck-corner equivalence. Sections 3–4 prove the relative interval-tilting theorem (Theorem 3.9) and its iterated staircase form (Theorem 4.5) from exact right Kan extensions and Rickard's derived Morita theorem. Proposition 5.4 identifies rational Dyck posets with coordinate staircases by an explicit bijection, and Proposition 5.5 plus Corollary 5.8 then give D^b(kDyck^above_{a,b}) ≃ D^b((B^st_{a,b})^op) without assuming the Chapoton–Ladkani–Rognerud conjecture. Theorem 6.6 is a chain of independent equivalences: Lemma 6.1 (tensor products of tilting complexes), Ladkani's replicated-algebra equivalence (43), Xing's equivalence (47), Lemma 6.5 (self-oppositeness), and Gottesman's Theorem E (49). None of these ingredients is the paper's own conclusion, none is by the present author, and the target equivalence is never used as a hypothesis. The identification of the new staircase corner with Xing's B0 is quoted from Xing [14, Proposition 4.33], and Xing's main equivalence is cited from [14, Theorem 4.5 and Proposition 4.25]; this is external reliance, not circularity. Lemma 6.5 is only sketched, and the finite checks in Section 8 stop before the Xing junction, so there are verification gaps and correctness risks, but no step reduces definitionally or statistically to its own input. There are no fitted parameters, no self-citation chain, and no renamed known result presented as a derivation. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central new results are proven internally. The main theorem additionally relies on several external results (Xing, Ladkani, Gottesman, Rickard) that are cited but not reproved. No free parameters or invented empirical entities appear; the mathematical objects introduced (Γ♯, C_j) are fully defined and their properties proven.

assumptions (7)
  • standard math Rickard's derived Morita theorem: a tilting complex T with End(T)^op ≅ B gives D^b(A) ≃ D^b(B).
    Invoked in Theorem 3.9 and Lemma 6.1 to convert endomorphism algebra isomorphisms into derived equivalences.
  • domain assumption Ladkani's replicated-algebra equivalence: for Gorenstein Λ, Λ ⊗ kA_r^→ ≃_der Λ^{(r)} [9, Cor 1.3].
    Used in (46) and Theorem 9.3 to replace a tensor product with a path algebra by a replicated algebra.
  • domain assumption Xing's theorem: for coprime a,b, B0^{(a+b)} ≃_der A^a_{b+1}, and B0 ≅ eA'e with the listed notation [14, Thm 4.5, Prop 4.25, Prop 4.33].
    This is the only coprimality-dependent step in the proof of the CLR conjecture (Theorem 6.6); it is cited without proof.
  • domain assumption Gottesman's Theorem E: kJ_{m,n} ≃_der A^{m-1}_{n+1} in his convention [5].
    Used in (49) to equate the path lattice algebra with a higher Auslander algebra.
  • domain assumption Index convention conversions among Jasso-Külshammer, Oppermann-Thomas, Xing, and Gottesman (Remark 2.8).
    Needed to equate A^a_{b+1} in different papers and with the Fukaya-categorical notation in Section 7.
  • standard math Finite global dimension implies Gorenstein for finite-dimensional algebras.
    Used in Section 9 to apply Ladkani's theorem to BΩ.
  • domain assumption Dyckerhoff-Jasso-Lekili and Di Dedda equivalences relating higher Auslander algebras, symmetric product Fukaya categories, and Fukaya-Seidel categories ([3], [2]).
    Used only in Section 7 for the geometric interpretation; not load-bearing for the algebraic proof of the CLR conjecture.

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Pith. "Pith review of Relative Interval Tilting, Higher Auslander Staircase Corners and Rational Dyck Posets." pith.science (2026). https://pith.science/paper/C3M6GV4I

@misc{pith2026260806696,
  author       = {Pith},
  title        = {Pith review of: Relative Interval Tilting, Higher Auslander Staircase Corners and Rational Dyck Posets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3M6GV4I}},
  note         = {Machine review of arXiv:2608.06696}
}
abstract

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton--Ladkani--Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton--Ladkani--Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya--Seidel categories of symmetric Brieskorn--Pham singularities.

Figures

Figures reproduced from arXiv: 2608.06696 by the authors.

Figure 1
Figure 1. The five below-diagonal rational (3, 4)-Dyck paths, labelled by their horizontal-step coordinates c(ℓ) and their images β(c(ℓ)). The β-images are the vertices of Ω3,4; its Hasse quiver is drawn in Section 8. Thus [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗

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Works this paper leans on

14 extracted references · 9 canonical work pages

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