MathAudit

Publication metadata

Journal citation
Moscow Mathematical Journal 3(3), 899–934 (2003)
Publication date
2003
arXiv counterpart
arXiv:math/0208033v2
MSC 2020
53D17, 14M15, 05E14, 05E16
As specified in the publication (MSC 2000)
53D17, 14M15, 05E15 ↗Final author-manuscript fallback, first pageThe printed 53D17 and 14M15 carry forward unchanged. In the official MSC2010-to-MSC2020 conversion, the printed 05E15 is redistributed to 05E14 and 05E16.

Abstract

The paper introduces Poisson structures and toric actions compatible with cluster algebra mutations and studies the symplectic-leaf and topological structure of the generic torus-orbit locus.

Role in dependence graphs

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Audit summary

Audited against an explicitly disclosed arXiv fallback

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Exact reviewed source

arXiv:math/0208033v2 · explicit fallback for inaccessible version of record

Michael Gekhtman, Michael Shapiro, Alek Vainshtein. Cluster algebras and Poisson geometry. arXiv:math/0208033v2, final version.

The Moscow Mathematical Journal version of record was not freely retrievable during the audit. This report reviews the current and final arXiv version 2, submitted November 25, 2003, and does not describe it as the publisher version.

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Generated August 27, 2026
01Statements4 reported findingsNo statement issues found

The classifications of compatible Poisson brackets and finite canonical local mutation data, the description of the generic symplectic-leaf locus and its connected components, and the Grassmannian applications are correct. The introduction overstates the parameter range of one component-count formula, but the formal Corollary 3.9 and Proposition 3.10 give the correct ranges and values.

Theorems 1.4 and 1.6No issue found

The compatible-bracket and mutation-data classifications are correct

Pages 6–12 · Theorems 1.4 and 1.6 · arXiv:math/0208033v2

Mutation of a log-canonical bracket forces the matrix identity between the exchange matrix and the bracket matrix, and full row rank gives the converse and the stated parameter count. For the inverse problem, reciprocity of the one-variable factors and the rank-two periodicity test exhaust the finite involutive cases; the two surviving families specialize to the ordinary cluster mutation formulas after rescaling.

Lemma 2.4 and Theorem 2.10No issue found

The generic-leaf and connected-component descriptions are supported

Pages 16–19 · Lemmas 2.4, 2.8–2.9 and Theorem 2.10 · arXiv:math/0208033v2

On every all-nonzero cluster chart, logarithmic coordinates make the Poisson tensor constant. In the degenerate case, integral null vectors give Casimir monomials and compatible toric flows transverse to their level sets; full row rank excludes a nonzero null vector with vanishing tropical component, while the positive diagonal choice makes the displayed transverse derivative nonzero. Over R\mathbb{R}, mutation changes sign octants by the stated transvections, so connected components are exactly the orbits of the generated finite group.

Theorem 3.5, Corollary 3.9, and Proposition 3.10No issue found

The Grassmannian leaf and component formulas have the correct formal scope

Pages 28–35 · Theorem 3.5, Corollary 3.9, and Proposition 3.10 · arXiv:math/0208033v2

The kernel calculation for the log-canonical matrix gives the stated gcd⁡(k,n)\gcd(k,n) parity alternative for maximal symplectic-leaf codimension. The E6E_6 orbit formula applies when k>3k>3 and n−k>3n-k>3, and the remaining G(2,n)G(2,n) and G(3,n)G(3,n) cases are supplied separately, with Grassmann duality covering n−k=2,3n-k=2,3. The cited orbit count in Theorem 3.8 matches the exact earlier theorem used.

Introduction parameter rangeSuspected minor formal issue

The opening component-count sentence omits the upper bound on kk

Page 2 · final paragraph describing the Grassmannian application · arXiv:math/0208033v2

The introduction says that the refined open Bruhat cell has 3⋅2n−13\cdot 2^{n-1} components whenever k≥3k\geq 3 and n≥7n\geq 7. The proved range is 3≤k≤n−33\leq k\leq n-3 (and it also includes k=3,n=6k=3,n=6): Corollary 3.9 treats k>3k>3 and n−k>3n-k>3, Proposition 3.10 treats k=3k=3, and duality treats n−k=3n-k=3. For k=n−2k=n-2, Proposition 3.10 and duality instead give (n−1)2n−2(n-1)2^{n-2}, so the unique local repair is to replace the introductory range by 3≤k≤n−33\leq k\leq n-3. The numbered results themselves are unaffected.

02Proofs3 reported findingsNo proof issues found

The mutation calculations, Casimir and toric-flow arguments, sign-orbit topology, and Grassmannian determinant computations are correct and complete. Two indices in the printed proof of Proposition 3.6 are typographical errors with uniquely determined corrections.

Sections 1–2No issue found in the reviewed proof

The Poisson and topological arguments are complete

Pages 3–19 · proofs of Theorems 1.4, 1.6, and 2.10 · arXiv:math/0208033v2

The block-matrix identities propagate log-canonicity across every mutation, and the rank-two periodicity analysis eliminates every additional rational local datum. The toric weights lie exactly in the exchange-matrix nullspace, the Casimir derivatives are nonzero in the required directions, and the octant adjacency criterion is equivalent to the mod-two transvections. No missing case or unresolved external input remains in these proof chains.

Sections 3.1–3.4No issue found in the reviewed proof

The Grassmannian construction and counts are complete

Pages 19–35 · proofs of Theorem 3.5, Propositions 3.6–3.7 and 3.10, and Corollary 3.9 · arXiv:math/0208033v2

The Sklyanin bracket descends to the Grassmannian chart, the factorization-coordinate matrix yields the announced Casimir basis, and Jacobi's determinantal identity makes every mutated coordinate regular. The directed-grid exchange matrix is compatible with the bracket, and the external finite-orbit formula together with the directly cited small-rank orbit counts yields the component formulas.

Proposition 3.6, cases (ii) and (v)Suspected typo

Two printed indices have unique harmless corrections

Page 31 · proof of Proposition 3.6 · arXiv:math/0208033v2

In case (ii), the printed α=i=j=k−n\alpha=i=j=k-n is incompatible with the case hypothesis i<ji<j; the row ranges in the displayed minors and m=n−km=n-k uniquely require α=i+j+k−n=i+j−m\alpha=i+j+k-n=i+j-m. In case (v), the left side is printed f1j′f'_{1j} although the case assumes i>1i>1 and the displayed minor depends on ii; it must be fij′f'_{ij}. Both corrected formulas remain polynomials in the chart coordinates, which is the only conclusion used.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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