Pipe dream complexes and triangulations of root polytopes belong together

TL;DR

This paper establishes a geometric connection between pipe dream complexes, root polytopes, and flow polytopes, linking algebraic, combinatorial, and geometric structures through triangulations and polynomial specializations.

Contribution

It demonstrates that pipe dream complexes can be realized as triangulations of root polytopes and connects Grothendieck polynomials to subdivision algebra and polytope structures.

Findings

Pipe dream complexes are realizable as triangulations of root polytope vertex figures.

Grothendieck polynomial specializes to the h-polynomial of pipe dream complexes.

Root polytopes are projections of flow polytopes, sharing subdivision algebra.

Abstract

In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...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.