Mirković-Vilonen polytopes and a quiver construction of crystal basis in type A (Q2917604)

Let \(B(\infty)\) be the crystal basis of the negative half of the quantum universal enveloping algebra associated with \(\mathfrak{sl}_{n+1}(\mathbb{C})\). The aim of the paper is to give explicit isomorphisms between the Lagrangian (or quiver) realization, the realization by using the theory of PBW basis, and the realization in terms of Mirkovi\(\acute{c}\)-Vilonen (MV) polytopes of \(B(\infty)\). A quiver theoretical description of MV polytopes is given in type \(A_n\). As a by-product, a new proof of the Anderson-Mirkovi\(\acute{c}\) conjecture on the explicit forms of the actions of lowering Kashiwara operators on the set of MV polytopes is given.