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

Abstract

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 Δi(λ), whose lattice points encode the character of the irreducible representation Vλ. The string polytope Δi(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 = SLn+1(ℂ), and present a sufficient condition on i such that the toric variety XΔi(λ) of the string polytope Δi(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 XΔ 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 Δi(λ) is integral for any dominant integral weight λ, which in particular implies that the anticanonical limit toric variety XΔ i(λP) 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 XΔ i(λ).