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

Abstract

Let G be a semisimple algebraic group over [Formula: see text]. For a reduced word [Formula: see text] of the longest element in the Weyl group of G and a dominant integral weight [Formula: see text], one can construct the string polytope [Formula: see text], whose lattice points encode the character of the irreducible representation [Formula: see text]. The string polytope [Formula: see text] is singular in general and combinatorics of string polytopes heavily depends on the choice of [Formula: see text]. In this paper, we study combinatorics of string polytopes when [Formula: see text], and present a sufficient condition on [Formula: see text] such that the toric variety [Formula: see text] of the string polytope [Formula: see text] has a small toric resolution. Indeed, when [Formula: see text] has small indices and [Formula: see text] is regular, we explicitly construct a small toric resolution of the toric variety [Formula: see text] using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when [Formula: see text]. As a byproduct, we show that if [Formula: see text] has small indices then [Formula: see text] is integral for any dominant integral weight [Formula: see text], which in particular implies that the anticanonical limit toric variety [Formula: see text] of a partial flag variety [Formula: see text] is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold [Formula: see text] and obtain a formula of the disk potential of the Lagrangian torus fibration on [Formula: see text] obtained from a flat toric degeneration of [Formula: see text] to the toric variety [Formula: see text].