Mirkovi\'c-Vilonen Polytopes and a quiver construction of crystal basis in type $A$

In the current paper, we give a quiver theoretical interpretation of Mirkovi'c-Vilonen polytopes in type $A_n$. As a by-product, we give a new proof of the Anderson-Mirkovi'c conjecture which describes the explicit forms of the actions of lowering Kashiwara operators on the set of Mirkovi'c-Vilonen polytopes.