Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

Abstract

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.