Positivity for cluster algebras

By Kyungyong Lee, Ralf Schiffler. 2013

164 citations recorded by the source corpus.

We prove the positivity conjecture for all skew-symmetric cluster algebras.

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

View Paper: https://www.semanticscholar.org/paper/4d70e22516e18cc81ef4e397bfae2bcd4025befb

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. Ilke Çanakçi
2. P. Hacking
3. S. Keel
4. Mark Gross
5. M. Kontsevich
6. Ben Davison
7. Dylan Rupel
8. Ralf Schiffler

In Geometry and Topology

1. Mark Gross
2. Ben Davison
3. P. Hacking
4. S. Keel
5. M. Kontsevich
6. Ilke Çanakçi
7. Dylan Rupel
8. Ralf Schiffler

In Statistical and Nonlinear Physics

1. Ben Davison
2. Ilke Çanakçi
3. Ralf Schiffler
4. P. Hacking
5. Mark Gross
6. Dylan Rupel
7. Max Glick
8. S. Keel

Citations in the corpus

• Canonical bases for cluster algebras (2014, methodology): “oefficients in the cluster variables of any given seed. This was conjectured by Fomin and Zelevinsky in their original paper [FZ02a]. Positivity was obtained independently in the skew-symmetric case by [LS13], by an entirely different argument. In our proof the positivity in (1) and (6) both come from ”
• Cluster algebras and continued fractions (2016, background): “iables x 1::::;x N by a process called mutation. It is known that each cluster variable is a Laurent polynomial in x 1;:::;x N with integer coecients [FZ1] and that these coecients are non-negative [LS]. We refer the reader to [FZ4] for the precise denition of cluster algebras. In this paper, we are”
• Positivity for quantum cluster algebras (2016, methodology): “levinsky in [BZ05]. Specialising the quantum cluster positivity conjecture at q1/2 = 1 we obtain the classical positivity conjecture of Fomin and Zelevinsky [FZ02], recently proved by Lee and Schiffler [LS15] in the skew-symmetric case, and then in the case of geometric type by Gross, Hacking, Keel an”
• Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings (2012, unknown): “e cluster variables are rational functions, but Fomin and Zelevinsky showed in [FZ1] that they are Laurent polynomials with integer coefcients. Moreover, these coecients are known to be non-negative [LS]. An important class of cluster algebras is given by cluster algebras of surface type [GSV, FG1, ”
• Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, background): “e cluster variables are rational functions, but Fomin and Zelevinsky showed in [FZ1] that they are Laurent polynomials with integer coecients. Moreover, these coecients are known to be non-negative [LS]. An important class of cluster algebras is given by cluster algebras of surface type [GSV, FG1, F”
• Introduction to Cluster Algebras (2018, methodology): “[38, 29] Let A(x, B̃) be a cluster algebra.”