Small toric resolutions of toric varieties of string polytopes with small indices

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. For a reduced word $\bf i$ of the longest element in the Weyl group of $G$ and a dominant integral weight $\lambda$, one can construct the string polytope $\Delta_{\bf i}(\lambda)$, whose lattice points encode the character of the irreducible representation $V_{\lambda}$. The string polytope $\Delta_{\bf i}(\lambda)$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $\mathbf i$. In this paper, we study combinatorics of string polytopes when $G = SL_{n+1}(\mathbb{C})$, and present a sufficient condition on $\mathbf i$ such that the toric variety $X_{\Delta_{\mathbf i}(\lambda)}$ of the string polytope $\Delta_{\mathbf i}(\lambda)$ has a small toric resolution. Indeed, when $\mathbf i$ has small indices and $\lambda$ is regular, we explicitly construct a small toric resolution of the toric variety $X_{\Delta_{\bf i}(\lambda)}$ using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when $n < 4$. As a byproduct, we show that if $\mathbf i$ has small indices then $\Delta_{\mathbf i}(\lambda)$ is integral for any dominant integral weight $\lambda$, which in particular implies that the anticanonical limit toric variety $X_{\Delta_{\bf i}(\lambda_P)}$ of a partial flag variety $G/P$ is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold $G/B$ and obtain a formula of the disk potential of the Lagrangian torus fibration on $G/B$ obtained from a flat toric degeneration of $G/B$ to the toric variety $X_{\Delta_{\bf i}(\lambda)}$.