2019/03/13 by Ricky Ini Liu, Karola Mészáros, Liu, Ricky Ini +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1903.05548
openalex publication_date 2019/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of \mathfrakgln(ℂ). The integer point transform of the Gelfand-Tsetlin polytope GT(λ) projects to the Schur function sλ. Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials \mathfrakSw corresponding to Grassmannian permutations. For any permutation w ∈ Sn with column-convex Rothe diagram, we construct a polytope Pw whose integer point transform projects to the Schubert polynomial \mathfrakSw. 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 \mathfrakSw for all w ∈ Sn. However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope Pw is a convex polytope. We also show that Pw 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.