On tropical dualities in cluster algebras

By Tomoki Nakanishi, Andrei Zelevinsky. 2011

167 citations recorded by the source corpus.

We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $\gg$-vectors parameterizing cluster variables, and the $\cc$-vectors parameterizing the coefficients. We prove two identities relating these vectors to each other. The proofs depend on the sign-coherence assumption for $\cc$-vectors that still remains unproved in full generality.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d0c40182e2614a24aaa4df552fe00bb14d3d1288

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. B. Keller
2. Dong Ping Yang
3. Laurent Demonet
4. Hugh Thomas
5. David E. Speyer
6. M. Kontsevich
7. P. Hacking
8. S. Keel

In Geometry and Topology

1. B. Keller
2. Laurent Demonet
3. Mark Gross
4. P. Hacking
5. S. Keel
6. Dong Ping Yang
7. M. Kontsevich
8. Hugh Thomas

In Statistical and Nonlinear Physics

1. Laurent Demonet
2. Dong Ping Yang
3. Hugh Thomas
4. B. Keller
5. David E. Speyer
6. P. Hacking
7. Mark Gross
8. S. Keel

Citations in the corpus

• Universal geometric cluster algebras (2012, unknown): “Again, the argument from [17] also proves a statement with weaker hypothesis and weaker conclusion:”
• A survey on maximal green sequences (2019, background): “As shown in [96], the theorem is central in the theory of cluster algebras.”
• Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “ is the case of finite-dimensional algebras of finite global dimension. The four concepts are closely related to important concepts in cluster theory such as clusters ([24]), c-matrices and g-matrices ([25, 56]) and cluster-tilting objects ([19]). In the finitedimensional setting, the first work in this”
• Donaldson-Thomas theory and cluster algebras (2010, background): “ed as a tropical counterpart of the xvariable, while the c-vector can be viewed as a tropical counterpart of the yvariable. The duality between the g- and the c-vectors is called toropical duality in [NZ] 9. From our view point, the x-variable corresponds to the “projective” Γi and the y-variable co”
• Canonical bases for cluster algebras (2014, methodology): “By [NZ], the sign coherence for c-vectors (proved in Corollary 5.”
• Acyclic Cluster Algebras Revisited (2012, methodology): “columns of the bottom half of Bev. It has recently been understood that the c-vectors play an important role in the behaviour of a cluster algebra associated to B0. Nakanishi and Zelevinsky showed in [NZ] that, once it is established that the c-vectors are sign-coherent, meaning that, for each c-vec”
• Ordered Exchange Graphs (2013, unknown): “For more details on c-vectors, we refer the reader to [35] where they were introduced and to [84, 83, 98, 82, 66] where they were studied.”