Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of $\mathfrak{ gl}_n(\mathbb{ C})$. The integer point transform of the Gelfand-Tsetlin polytope $\mathrm{ GT}(\lambda)$ projects to the Schur function $s_{ \lambda}$. Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials $\mathfrak{ S}_{ w}$ corresponding to Grassmannian permutations. For any permutation $w \in S_n$ with column-convex Rothe diagram, we construct a polytope $\mathcal{ P}_{ w}$ whose integer point transform projects to the Schubert polynomial $\mathfrak{ S}_{ w}$. Such a construction has been sought after at least since the construction of twisted cubes by Grossberg and Karshon in 1994, whose integer point transforms project to Schubert polynomials $\mathfrak{ S}_{ w}$ for all $w \in S_n$. However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope $\mathcal{ P}_{ w}$ is a convex polytope. We also show that $\mathcal{ P}_{ w}$ is a Minkowski sum of Gelfand-Tsetlin polytopes of varying sizes. When the permutation $w$ is Grassmannian, the Gelfand-Tsetlin polytope is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a flow polytope.
Categories: math.CO