2019/12/02 by Yunhyung Cho, Yoosik Kim, Cho, Yunhyung +5
Arts and Humanities · Mathematics · #14E15 #14M15 #14M25 (Primary) #52B20 #53D37 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Medieval Literature and History #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1912.00658
openalex publication_date 2019/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a semisimple algebraic group over ℂ. For a reduced word \bf i of the longest element in the Weyl group of G and a dominant integral weight λ, one can construct the string polytope Δ\bf i(λ), whose lattice points encode the character of the irreducible representation Vλ. The string polytope Δ\bf i(λ) 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 = SLn+1(ℂ), and present a sufficient condition on \mathbf i such that the toric variety X_Δ\mathbf i(λ) of the string polytope Δ\mathbf i(λ) has a small toric resolution. Indeed, when \mathbf i has small indices and λ is regular, we explicitly construct a small toric resolution of the toric variety X_Δ\bf 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 \mathbf i has small indices then Δ\mathbf i(λ) is integral for any dominant integral weight λ, which in particular implies that the anticanonical limit toric variety X_Δ\bf 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 X_Δ\bf i(λ).