Yuki Kanakubo - Inequalities defining polyhedral realizations and monomial realizations (...)

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

Yuki Kanakubo (Ibaraki University)

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