Resumen de Polyhedral realizations of crystal bases and convex-geometric Demazure operators

Naoki Fujita

• The main object in this paper is a family of rational convex polytopes whose lattice points give a polyhedral realization of a highest weight crystal basis. Every polytope in this family is identical to a Newton–Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of polytopes in this family is given by Kiritchenko’s Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.