h-polynomials via reduced forms

The flow polytope F-(G) over tilde is the set of nonnegative unit flows on the graph (G) over tilde. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope F-(G) over tilde into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial Pi((i,j)is an element of E(G))x(ij). We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of h -polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi -classical Yang -Baxter algebra.