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

Let G be a semisimple algebraic group over ℂ. For a reduced word i of the longest element in the Weyl group of G and a dominant integral weight λ , one can construct the string polytope Δ i (λ) , whose lattice points encode the character of the irreducible representation V λ . The string polytope Δ i (λ) is singular in general and combinatorics of string polytopes heavily depends on the choice of i. In this paper, we study combinatorics of string polytopes when G = S L n + 1 (ℂ) , and present a sufficient condition on i such that the toric variety X Δ i (λ) of the string polytope Δ i (λ) has a small toric resolution. Indeed, when i  has small indices and λ is regular, we explicitly construct a small toric resolution of the toric variety X Δ i (λ) 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 i has small indices then Δ i (λ) is integral for any dominant integral weight λ , which in particular implies that the anticanonical limit toric variety X Δ i (λ 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 Δ i (λ) . [ABSTRACT FROM AUTHOR]