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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 $B^{u,v}$ is a double Bruhat cell.

On the other hand, in Grassmannian case (correspond to type A case), from any Postnikov diagram, in the paper by Scott, an initial seed is constructed.

In Grassmannian case, what are the relations between the initial seeds constructed by Berenstein-Fomin-Zelevinsky and the initial seed from Postnikov diagrams? Thank you very much.

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  • $\begingroup$ The most direct relationship is between the Grassmannian and its big Schubert cell, where the coordinate rings are related by looking at a degree 0 subalgebra of a certain (small) localisation of the Grassmannian. This translates into removing a frozen variable from the initial quiver of the Grassmannian seed, but the actual variables change in a much more complicated way. Apologies for the slightly bad form in saying this, but you might find some of my papers on this with Launois helpful, and/or contact me by email if you like. (If you prefer, just ignore the word "quantum" in those papers.) $\endgroup$
    – Jan Grabowski
    Commented Jan 5, 2021 at 12:00
  • $\begingroup$ @Jan Grabowski, thank you very much! $\endgroup$
    – Jianrong Li
    Commented Jan 5, 2021 at 14:35

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