Abstract
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 $\beta$-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 $\beta$-Grothendieck polynomials.
Highlights
In this paper we provide geometric realizations of pipe dream complexes PD (π) of permutations π = 1π, where π is a dominant permutation on 2, 3, . . . , n as well as the subword complexes that are the cores of the pipe dream complexes PD (π)
Subword complexes are simplicial complexes introduced by Knutson and Miller in [10, 11] to illustrate the combinatorics of Schubert polynomials and determinantal ideals
Subword complexes were first shown to relate to triangulations of root polytopes by Mészáros in [17], where the author gives a geometric realization of the pipe dream complex of [1, n, n − 1, . . . , 2] and whose work served as the stepping stone for the present project
Summary
In this paper we provide geometric realizations of pipe dream complexes PD (π) of permutations π = 1π , where π is a dominant permutation on 2, 3, . We realize PD (π) as (repeated cones of) regular triangulations of the root polytopes P(T (π)). Subword complexes were first shown to relate to triangulations of root polytopes by Mészáros in [17], where the author gives a geometric realization of the pipe dream complex of [1, n, n − 1, .
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