On the combinatorics of string polytopes

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