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Results tagged with cluster-algebras
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user 144655
Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
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Geometric description of a type $A$ cluster algebra with universal coefficients
In Thm. 12.4 in Fomin and Zelevinsky - Cluster algebras IV: Coefficients we are given a recipe for constructing a cluster algebra with universal coefficients. The recipe is given in terms of (almost p …
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Cluster algebras and tropical points
Recently, I have been looking at some articles about bases for cluster algebras and came across the idea of tropical points. I should highlight here that unfortunately I have no background on algebrai …
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Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations
Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary choi …
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Piecewise linear $\sigma_i$ - notation question
In cluster algebra framework, in order to get root clusters, a modified version of a simple reflection is used. Define $\sigma_i:\Phi_{\geq -1} \to \Phi_{\geq -1}$ by setting:
$ \sigma_i(\alpha) =
…
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Choice of a ground ring for cluster algebras
In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . Som …
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About cluster variables obtained by (sequentially) mutating at exchangeable variables from a...
Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a sequenc …
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