Linear independence of cluster monomials for skew-symmetric cluster algebras

By Giovanni Cerulli Irelli, B. Keller, Daniel Labardini-Fragoso, Pierre-Guy Plamondon. Compositio Mathematica, 2012

99 citations recorded by the source corpus.

Abstract Fomin–Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ca51b01ca1d9cebcbe6f2d82c90d03b98219b7da

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In Algebra and Number Theory

1. P. Hacking
2. Thomas Brustle
3. S. Keel
4. Mark Gross
5. M. Kontsevich
6. Pavlo Pylyavskyy
7. Myungho Kim
8. Dong Yang

In Discrete Mathematics and Combinatorics

1. P. Hacking
2. M. Kontsevich
3. Pavlo Pylyavskyy
4. S. Keel
5. Thomas Brustle
6. Mark Gross
7. Masaki Kashiwara
8. Dong Yang

In Geometry and Topology

1. Mark Gross
2. P. Hacking
3. M. Kontsevich
4. S. Keel
5. T. Bridgeland
6. I. Smith
7. B. Keller
8. Thomas Brustle

Citations in the corpus

• Ordered Exchange Graphs (2013, background): “3], the cluster algebras with principal coefficients govern the combinatorics of all cluster algebras (see [23] for a stronger result in the skew-symmetric case).”
• Tensor diagrams and cluster algebras (2012, background): “existence of a basis B satisfying a small subset of theconditions in Conjecture 9.5. Forexample, the existence ofabasis B satisfying (1) is equivalent to linear independence of cluster monomials, cf. [5]. Perhaps even more important is the problem of determining a cluster structure in a given ring w”
• Quadratic differentials as stability conditions (2013, background): “ons of the surface (S,M) and t-structures in the corresponding category D(S,M). Theorem 9.8. There are isomorphisms of graphs Tri⊲⊳(S,M) ∼=Exch△(D), Tri⋄ ⊲⊳(S,M) ∼=Exch⋄ △(D). Proof. Corollary 3.6 of [6] and Theorem 5.6 of [22] together imply that the heart exchange graph Exch△(D) is isomorphic to t”
• Canonical bases for cluster algebras (2014, methodology): “ut the convexity assumption). Note (7) includes the linear independence of cluster monomials, which has already been established (without convexity assumptions) for skew-symmetric cluster algebras in [CKLP], by a very different argument. The linear independence of cluster monomials in the principal c”
• Monoidal categorification of cluster algebras (2014, background): “Thus, with a result of [3], one can expect that every cluster monomial of Aq(n(w)) is contained in the upper global basis B(w), which is named the quantization conjecture by Kimura ([15]).”