  In this paper we investigate the idea of a tropical critical point of the
superpotential for the full flag variety of type A. Recall that associated to
an irreducible representation of G=SLn(C) are various polytopes whose integral
points parameterize a basis for the representation, e.g. the Gelfand-Zetlin
polytope. Such polytopes can be constructed via the theory of geometric
crystals by tropicalising a certain function, and in fact, the function
involved coincides with the superpotential from the Landau-Ginzburg model for
G/B coming from mirror symmetry. In mirror symmetry a special role is played by
the critical points of the superpotential, and motivated by this, we give a
definition of the tropical critical points and use it to find a canonical point
in each polytope. We then characterize the highest weights for which this
tropical critical point is integral and therefore corresponds to a basis vector
of the corresponding representation. Finally we give an interpretation of the
tropical critical point by constructing a special vector in the representation
using Borel-Weil theory and conjecturing a correspondence between this vector
and the tropical critical point.
