# Geometric description of a kind \$A\$ cluster algebra with common coefficients Answer

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Geometric description of a kind \$A\$ cluster algebra with common coefficients

In $$textual content{Thm.} ; 12.4$$ in Fomin’s and Zelevinsky’s paper on coefficients we’re given a recipe for establishing a cluster algebra with common coefficients. The recipe is given by way of (virtually optimistic) coroots and I’m making an attempt to grasp how this interprets into the habitual cluster algebraic kind A setup moving convex polygons,triangulations, and so forth. If I grasp accurately, because the preliminary seed we take any zig-zag triangulation. The coefficient bailiwick is the Tropical semifield (freely) generated by twin diagonals, I cerebrate. For occasion, in $$A_2$$, we would have that $$mathbb{P}=textual content{Trop}(x_{12}x_{34},x_{12}x_{45},x_{23}x_{45},x_{23}x_{15},x_{34}x_{15})$$.In different phrases, every generator represents a pair of “disjoint” sides of a pentagon. Now, I’m not positive in regards to the subsequent sever. It might be one thing simple however I’m failing to behold the geometric description of the $$Y$$-coefficients within the preliminary coefficient tuple ($$12.5$$ sever of $$textual content{Thm.} ; 12.5$$). Any ameliorate would breathe very mighty appreciated.

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