Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $y$.

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:

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.

share|improve this question

2 Answers 2

I believe this is conjectured to be true in general.

As for cluster algebras from surfaces, maybe the answer lies in http://math.berkeley.edu/~williams/papers/MSW-July24v5.pdf ?

share|improve this answer
The reference for the conjecture is Conjecture 7.6 of Cluster Algebras IV by Fomin and Zelevinsky. –  Nathan Reading Apr 24 at 20:45

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 (preprint).

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.