8 research outputs found
h-Polynomials of Reduction Trees
Reduction trees are a way of encoding a substitution procedure dictated by
the relations of an algebra. We use reduction trees in the subdivision algebra
to construct canonical triangulations of flow polytopes which are shellable. We
explain how a shelling of the canonical triangulation can be read off from the
corresponding reduction tree in the subdivision algebra. We then introduce the
notion of shellable reduction trees in the subdivision and related algebras and
define h-polynomials of reduction trees. In the case of the subdivision
algebra, the h-polynomials of the canonical triangulations of flow polytopes
equal the h-polynomials of the corresponding reduction trees, which motivated
our definition. We show that the reduced forms in various algebras, which can
be read off from the leaves of the reduction trees, specialize to the shifted
h-polynomials of the corresponding reduction trees. This yields a technique for
proving nonnegativity properties of reduced forms. As a corollary we settle a
conjecture of A.N. Kirillov.Comment: 23 pages, 3 figure
Subword complexes via triangulations of root polytopes
Subword complexes are simplicial complexes introduced by Knutson and Miller
to illustrate the combinatorics of Schubert polynomials and determinantal
ideals. They proved that any subword complex is homeomorphic to a ball or a
sphere and asked about their geometric realizations. We show that a family of
subword complexes can be realized geometrically via regular triangulations of
root polytopes. This implies that a family of -Grothendieck polynomials
are special cases of reduced forms in the subdivision algebra of root
polytopes. We can also write the volume and Ehrhart series of root polytopes in
terms of -Grothendieck polynomials.Comment: 17 pages, 15 figure
Toric matrix Schubert varieties and root polytopes (extended abstract)
International audienceStart with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations
Toric matrix Schubert varieties and root polytopes (extended abstract)
Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations