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, namely it 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.Mathematics Subject Classifications: 05E05Keywords: Schubert polynomials, Gelfand-Tsetlin polytopes, flow polytopes