$h$-Polynomials of Reduction Trees

Abstract

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.