Schubert polynomials as projections of Minkowski sums of GelfandTsetlin polytopes
Abstract
GelfandTsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of $\mathfrak{gl}_n(\mathbb{C})$. The integer point transform of the GelfandTsetlin 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 columnconvex 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 GelfandTsetlin polytopes of varying sizes. When the permutation $w$ is Grassmannian, the GelfandTsetlin polytope is recovered. We conclude by showing that the GelfandTsetlin polytope is a flow polytope.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.05548
 Bibcode:
 2019arXiv190305548I
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 17 pages, 5 figures