h-Polynomials of Reduction Trees

TL;DR

This paper introduces h-polynomials of reduction trees within the subdivision algebra, linking them to shellable triangulations of flow polytopes and providing a new method to prove nonnegativity properties, settling a conjecture of Kirillov.

Contribution

It defines h-polynomials for reduction trees, connects them to shellable triangulations, and uses this framework to prove nonnegativity properties and settle a conjecture.

Findings

h-polynomials of reduction trees match those of canonical triangulations

reduced forms specialize to shifted h-polynomials

technique established for proving nonnegativity of reduced forms

Abstract

Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. 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 A.N. Kirillov.