Abstract. We introduce bubbling diagrams and show that they compute the support of the Grothendieck polynomial of any vexillary permutation. Using these diagrams, we show that the support of the top homogeneous component of such a Grothendieck polynomial coincides with the support of the dual character of an explicit flagged Weyl module. We also show that the homogenized Grothendieck polynomial of a vexillary permutation has M-convex support.

Gelfand--Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand--Tsetlin polytope ${GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well known that the Gelfand--Tsetlin polytope is both a marked order polytope and a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes.

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.

We develop a method of proving nonnegativity of the coefficients of certain polynomials, also called reduced forms, defined by Kirillov in his quasi-classical Yang--Baxter algebra, its abelianization, and related algebras. It has been shown previously that the relations of the abelianization of the quasi-classical Yang--Baxter algebra, also called the subdivision algebra, encode ways of subdividing flow polytopes. In turn, these subdivisions can be represented as reduced forms, or as reduction trees. 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 Kirillov.