Pipe Dream Complexes and Triangulations of Root Polytopes Belong Together

Abstract

We show that the pipe dream complex associated to the permutation $1\text{ } n \text{ }n-1\text{ } \cdots \text{ }2$ can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the $h$-polynomial of the corresponding pipe dream complex, which in certain cases equals the $h$-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to reduced forms in subdivision algebras and root (and flow) polytopes. We also show that root polytopes can be seen as projections of flow polytopes, explaining that these families of polytopes possess the same subdivision algebra.