On the combinatorics of string polytopes

For a reduced word ${\bf i}$ of the longest element in the Weyl group of $\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$ which parametrizes the dual canonical bases. In this paper, we classify all ${\bf i}$'s such that $C_{\bf i}$ is simplicial. We also prove that for any regular dominant weight $\lambda$ of $\mathfrak{sl}_{n+1}(\mathbb{C})$, the corresponding string polytope $\Delta_{\bf i}(\lambda)$ is unimodularly equivalent to the Gelfand-Cetlin polytope associated to $\lambda$ if and only if $C_{\bf i}$ is simplicial. Thus we completely characterize Gelfand-Cetlin type string polytopes in terms of ${\bf i}$.