Gelfand–Tsetlin polytopes and the integer decomposition property

Let PP be the Gelfand–Tsetlin polytope defined by the skew shape λ/μ and weight w. In the case corresponding to a standard Young tableau, we completely characterize for which shapes λ/μ the polytope PP is integral. Furthermore, we show that PP is a compressed polytope whenever it is integral and corresponds to a standard Young tableau. We conjecture that a similar property holds for arbitrary w, namely that PP has the integer decomposition property whenever it is integral.Finally, a natural partial ordering on GT-polytopes is introduced that provides information about integrality and the integer decomposition property, which implies the conjecture for certain shapes.