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In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have cluster algebra structure on $Gr(2,3)$ and $Gr(2,4)$? Thank you very much.

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    $\begingroup$ $Gr(k,n) \cong Gr(n-k,n)$. $\endgroup$
    – Sasha
    Commented Sep 14, 2016 at 14:55

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The canonical source for the cluster structure on (all) Grassmannians is the aptly-titled

Joshua S. Scott, MR 2205721 Grassmannians and cluster algebras, Proc. London Math. Soc. (3) 92 (2006), no. 2, 345--380.

More information can also be found in papers citing this one.

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