A conjecture on monomial realizations and polyhedral realizations for crystal bases

Crystal bases are powerful combinatorial tools in the representation theory of quantum groups U q ( 𝔤 ) for a symmetrizable Kac-Moody algebras 𝔤 . The polyhedral realizations are combinatorial descriptions of the crystal base B ( ∞ ) for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when 𝔤 is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases B ( λ ) of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when 𝔤 is a classical Lie algebra, a rank 2 Kac-Moody algebra or a nonexceptional affine Lie algebra.