Published paper
Cluster algebras III: Upper bounds and double Bruhat cells
MSC 2020
16S9905E1405E1614M1722E46
Abstract
The paper introduces upper cluster algebras as intersections of Laurent polynomial rings, proves mutation invariance of upper bounds under coprimality, identifies acyclic cluster algebras with their upper counterparts, and realizes coordinate rings of double Bruhat cells as upper cluster algebras.
Role in dependence graphs
Proof-critical source
On Lagrangianity of -supports of holonomic -modules and --modules
This paper is included only for the following marked statements:
- Theorem 1.5 · Duke statement p. 5; final-manuscript proof pp. 25–28Mutation invariance of the commutative upper bound for coprime seeds.
- Corollary 1.7 and Proposition 1.8 · Duke pp. 5–6; final-manuscript proof of Proposition 1.8 pp. 23–25A totally coprime upper cluster algebra equals one finite upper bound; full-rank extended exchange matrices make every seed coprime.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:math/0305434v3 · explicit fallback for the inaccessible Duke version of record
Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Cluster algebras III: Upper bounds and double Bruhat cells. Duke Mathematical Journal 126 (2005), no. 1, 1–52. Reviewed in final arXiv:math/0305434v3 because the Duke version of record was not publicly downloadable.
The exact Duke Mathematical Journal version of record was subscription-gated during this audit. The current and final arXiv version 3, also available as the authors' final manuscript, was reviewed as the explicit proof-bearing fallback; this report does not claim inspection of the publisher PDF.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The reviewed final manuscript states the mutation-invariance, total-coprimality, acyclic-presentation, finite-generation, and double-Bruhat-cell results at the hypotheses established in the paper. The proof-critical statements used by Berenstein–Zelevinsky and by Muller–Nguyen–Trampel–Yakimov are correct.
Theorem 1.5 and Corollary 1.7No issue found
Coprime mutation preserves the upper bound
arXiv:math/0305434v3, manuscript pages 4–5 and 25–28
Theorem 1.5 requires both adjacent seeds to be coprime and proves equality of their finite upper bounds. Corollary 1.7 correctly upgrades this to the upper cluster algebra when every mutation-equivalent seed is coprime. The intersection calculation in Section 4 treats the zero-interaction and rank-two interaction cases separately, and Lemma 4.6 completes mutation invariance without invoking the older Laurent-phenomenon theorem.
Final arXiv version 3 ↗Proposition 1.8No issue found
Full rank forces total coprimality
arXiv:math/0305434v3, manuscript pages 5 and 23–25
Lemma 3.1 identifies non-coprime exchange polynomials with columns proportional by a ratio of odd coprime integers. Lemma 3.2 factors a matrix mutation through invertible integer matrices, so rank is mutation-invariant. A full-rank extended exchange matrix therefore has no proportional pair of columns in any seed, which proves Proposition 1.8 and supplies the exact total-coprimality input used in the dependency graph.
Final arXiv version 3 ↗Theorems 1.16, 1.18, and 1.20No issue found
The acyclic lower and upper descriptions have the stated scope
arXiv:math/0305434v3, manuscript pages 6–7 and Sections 5–7
The lexicographic leading-term argument proves independence of standard monomials for acyclic seeds, while an oriented-cycle identity proves the converse. The induction in Section 6 uses coprimality exactly where saturation by the first exchange polynomial is required. Section 7 then removes coprimality for the equality of the cluster algebra with the lower bound by specializing from universal coefficients. The resulting hypotheses match the statements.
Final arXiv version 3 ↗Theorem 2.10Correct with explicit imported inputs
The double-Bruhat-cell theorem is separated from the self-contained upper-bound core
arXiv:math/0305434v3, manuscript pages 16–18
Once Lemmas 2.12 and 2.13 provide the toric charts and the codimension-two complement, the regular-function intersection proves the coordinate-ring identification. Those two lemmas explicitly import generalized-minor and twist results from the cited earlier papers rather than silently treating them as internal. They are not ancestors of the focal graph's uses of Theorem 1.5, Corollary 1.7, or Proposition 1.8, so this ordinary audit records but does not recursively re-audit that separate Lie-theoretic source chain.
Final arXiv version 3 ↗02Proofs3 reported findingsNo proof issues found
The upper-bound and total-coprimality proofs needed by the dependency graph are complete in the reviewed final manuscript. The longer acyclic arguments were traced through their induction and saturation steps; no proof-breaking gap was found. The double-Bruhat branch clearly identifies its external generalized-minor inputs.
Section 4No issue found in the reviewed proof
The rank-two divisibility calculation closes the marked source chain
arXiv:math/0305434v3, manuscript pages 25–28 · Proposition 4.3 and Lemmas 4.4–4.6
After freezing all but two mutable variables, coprimality converts simultaneous divisibility into divisibility by the product in the disconnected case. In the interacting case, Lemmas 4.4 and 4.5 rewrite the intersection in the four generators. The final exchange relation and coefficient identity place the twice-mutated variable in the same ring. These are the commutative steps that the quantum proof later compresses, and they are valid in the form imported there.
Final arXiv version 3 ↗Section 3No issue found in the reviewed proof
The rank argument is reproved rather than inherited
arXiv:math/0305434v3, manuscript pages 23–25 · Lemmas 3.1 and 3.2
Although Lemma 3.2 notes an earlier appearance, the manuscript supplies the factorization of the mutated matrix by two invertible matrices. Proposition 1.8 therefore does not depend on an unaudited paper for its proof-critical rank-preservation step.
Final arXiv version 3 ↗Exact-form coverageExplicit final-manuscript fallback
The publisher file was not available for comparison
Source provenance for Duke Mathematical Journal 126 (2005), 1–52
The audit used the current and final arXiv version 3 and its source. The Duke version of record was not freely downloadable, so publisher-only copyediting or pagination changes cannot be certified here. No mathematical discrepancy was found in the final public manuscript, and the report does not label that file as the publisher version of record.
Duke article landing page ↗03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.