On combinatorics of string polytopes in types B and C

Abstract

A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric information of a flag variety such as toric degenerations, Newton–Okounkov bodies, mirror symmetry, Schubert calculus, and so on. In this paper, we study combinatorial properties of string polytopes in types B and C by giving an explicit description of string cones in these types which is analogous to Gleizer–Postnikov's description of string cones in type A. As an application, we characterize string polytopes in type C which are unimodularly equivalent to the Gelfand–Tsetlin polytope in type C for a specific highest weight. © 2025 Elsevier Ltd