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In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u,...

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$. We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...