On the combinatorics of string polytopes

Abstract

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