Do Denominator Vectors Determine the Cluster Variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:50:03Z http://mathoverflow.net/feeds/question/75411 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75411/do-denominator-vectors-determine-the-cluster-variable Do Denominator Vectors Determine the Cluster Variable Steve 2011-09-14T15:13:50Z 2011-09-27T14:49:05Z <p>Given a cluster algebra $A=A(\mathbf{x},Q)$, the Laurent Phenomenon states that all the cluster variables of $A$ are Laurent polynomials in the elements of $\mathbf{x}$. Thus, any cluster variable $y$ can be written $$y=\frac{p(x_1,\dots,x_n)}{x_1^{d_1}\cdots x_n^{d_n}}$$ where $p$ is a polynomial and $d_i$ are positive integers. We call $d(y):=(d_1,\dots,d_n)$ the denominator vector of $y$.</p> <p>If $Q$ is mutation equivalent to a simply laced Dynkin diagram, all cluster variables are uniquely determined by their denominator vector. I would like to know to what extent this holds in general. That is:</p> <p>Is it true that for any cluster algebra, the clusters are determined by their denominator vectors? If not, what classes of cluster algebras have this property? I am particularly interested in surface cluster algebras.</p> http://mathoverflow.net/questions/75411/do-denominator-vectors-determine-the-cluster-variable/75628#75628 Answer by Sam Clearman for Do Denominator Vectors Determine the Cluster Variable Sam Clearman 2011-09-16T17:59:01Z 2011-09-16T19:15:34Z <p>I believe this is conjectured to be true in general.</p> <p>As for cluster algebras from surfaces, maybe the answer lies in <a href="http://math.berkeley.edu/~williams/papers/MSW-July24v5.pdf" rel="nofollow">http://math.berkeley.edu/~williams/papers/MSW-July24v5.pdf</a> ?</p> http://mathoverflow.net/questions/75411/do-denominator-vectors-determine-the-cluster-variable/76519#76519 Answer by Julian Kuelshammer for Do Denominator Vectors Determine the Cluster Variable Julian Kuelshammer 2011-09-27T14:49:05Z 2011-09-27T14:49:05Z <p>For acyclic cluster algebras, a theorem like that can be found in "Buan, Marsh, Reiten: Denominators of cluster variables, J. London Math. Soc. 2009", Theorem 1.3 <a href="http://www1.maths.leeds.ac.uk/~marsh/research_articles/pp27.pdf" rel="nofollow">(preprint)</a>. </p>