Geometric crystals and cluster varieties

The notion of geometric crystal was initiated by A.Berenstein and D.Kazhdan to consider certain geometric analogue to the Kashiwara's crystal base theory. Positive geometric crystals can be tranfered to crystal (base) by considering tropicalization/ultra-discretization procedures. A pair of cluster variety $(A,X)$, called "cluster ensemble" is defined by glueing algebraic tori using the "mutations of cluster algebra", which is introduced by V.Fock and A.Goncharov. We shall define the geometric crystal structure on them and show some compatibility among them in the classical type $A$ case.