Orateur
Description
Crystal bases B(∞), B(λ) 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(λ) or B(∞). 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(∞) 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(λ) 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.
