TiTle:  A conjecture on inequalities defining polyhedral realizations and monomial realizations of crystal bases

Abstract:  The quantum groups introduced by Drinfeld and Jimbo are deformations of the universal enveloping algebras of Kac-Moody algebras. To study representations of Lie algebras or quantum groups, the crystal bases are powerful tools. We obtain several essential information of integrable highest weight representations or Verma modules from them. The crystal bases have a bunch of realizations via combinatorial objects. The polyhedral realizations invented by Nakashima-Zelevinsky describe a crystal base for the Verma module in terms of the set of integer points of a convex cone, which coincides with the string cone when the associated Lie algebra is finite dimensional simple. It is a fundamental problem to find an explicit form of this convex cone. On the other hand, the monomial realizations introduced by Kashiwara and Nakajima describe crystal bases via Laurent monomials in double indexed variables. In this talk, we give a conjecture that the inequalities defining the cone of polyhedral realizations can be obtained from monomial realizations for fundamental representations via tropicalizations. So far, it is shown that the conjecture is true in case of the Kac-Moody algebra is classical type or classical affine type.