Appears in collection : Combinatorics and Arithmetic for Physics - 2024

Crystal bases $B(\infy)$, $B(\lambda)$ are powerful tools to study representations of Lie algebras and quantum groups. We can get several essential information of integrable highest weight representations or Verma modules from $B(\lambda)$ or $B(\infty)$. To obtain such information from crystal bases, we need to describe them by combinatorial objects. The polyhedral realizations invented by Nakashima-Zelevinsky are combinatorial descriptions for $B(\infty)$ 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 and natural problem to find an explicit form of this convex cone. The monomial realizations introduced by Kashiwara and Nakajima are combinatorial expressions of crystal bases $B(\lambda)$ as Laurent monomials in double indexed variables. In this talk, we give a conjecture that the inequalities defining the cone of polyhedral realizations can be expressed in terms of monomial realizations of fundamental representations