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 pol...
['Karola Mészáros']
$h$-Polynomials of Reduction Trees