On the combinatorics of string polytopes

For a reduced word i of the longest element in Weyl group SLn+1(C), one can associate string cone Ci which parametrizes dual canonical bases. In this paper, we classify all i's such that is simplicial. We also prove for any regular dominant weight λ sln+1(C), corresponding polytope Δi(λ) unimodularly equivalent to Gelfand–Cetlin associated if and only Thus completely characterize type polytopes terms i.