Tropical multiwords and cluster structures on Plucker algebras

A word is a subset of a finite alphabet. A multiword is a collection of words. We identify multiwords with integer arrays. Arrays form a cone and there is a bijection between  integer points of
this cone  and the set of semi-standard Young tableaux. We endow arrays with a structure of tropical semiring. There is a structure of cluster algebra on the Plucker algebra (cluster algebras were
introduced by S. Fomin and A. Zelevinsky). We 'tropicalize'  the Plucker algebra and prove that 'tropicalization' of the cluster monoms are multiwords. We prove (modulo positivity conjecture) that cones of 'tropicalized' seeds define a fan in the cone of arrays.
For cluster algebras of finite type, such a fan has finitely many cones and their union coincide with the whole cone of arrays. This is a joint work with V.Danilov and A.Karzanov