[Paper Review] h-Polynomials via Reduced Forms

Karola Mészáros|arXiv (Cornell University)|Jul 10, 2014

Advanced Combinatorial Mathematics8 references4 citations

TL;DR

This paper establishes that reduced forms in the subdivision algebra of flow polytopes encode regular flag triangulations, generalizing h-polynomials of these triangulations. It proves that specialized reduced forms yield nonnegative polynomials, confirming a special case of Kirillov's conjecture on nonnegativity in the quasi-classical Yang-Baxter algebra via geometric and Ehrhart-theoretic methods.

ABSTRACT

The flow polytope $\mathcal{F}_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $\mathcal{F}_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. 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.

Motivation & Objective

• To establish a geometric interpretation of reduced forms in the subdivision algebra as h-polynomials of regular flag triangulations of flow polytopes.
• To prove that coefficients of reduced forms are nonnegative by linking them to triangulations and Ehrhart theory.
• To verify a special case of Kirillov's conjecture on nonnegativity of reduced forms in the quasi-classical Yang-Baxter algebra.
• To generalize known results on Ehrhart series and Kostant partition functions using reduced forms.

Proposed method

• Use the subdivision algebra to encode dissections of flow polytopes into simplices via reduction trees.
• Define a specific reduction order σ to construct coherent routes and cliques that yield regular flag triangulations.
• Show that the reduced form of a monomial corresponds to the h-polynomial of the triangulation when certain variables are specialized to 1.
• Apply the regularity of the triangulation to prove nonnegativity of coefficients in the reduced form.
• Relate the Ehrhart series of flow polytopes to reduced forms via the h-polynomial identity.
• Express the Ehrhart series in terms of Kostant partition functions using the identity i(F_G, m) = K_{G}(m, 0, ..., 0, -m).

Experimental results

Research questions

• RQ1 Can reduced forms in the subdivision algebra be interpreted as h-polynomials of triangulations of flow polytopes?
• RQ2 Does the regular flag triangulation of a flow polytope constructed via the subdivision algebra yield nonnegative coefficients in the reduced form?
• RQ3 Is the reduced form of the monomial x_{12}x_{23}...x_{n-1,n} in the quasi-classical Yang-Baxter algebra nonnegative when evaluated at x=(1,...,1) and β−1?
• RQ4 How are the Ehrhart series of flow polytopes related to reduced forms in the subdivision algebra?
• RQ5 Can the Kostant partition function be used to express the Ehrhart series of flow polytopes via reduced forms?

Key findings

• The reduced form of a monomial in the subdivision algebra, when specialized by setting certain variables to 1, equals the shifted h-polynomial of a regular flag triangulation of the corresponding flow polytope.
• The coefficients of the reduced form are nonnegative because they count faces in a regular flag triangulation.
• The reduced form of x_{12}x_{23}...x_{n-1,n} evaluated at x=(1,...,1) and β−1 in the quasi-classical Yang-Baxter algebra is a polynomial in β with nonnegative coefficients, confirming a special case of Kirillov's conjecture.
• The Ehrhart series of a flow polytope F_{G̃} is expressed as Q_G^{S(β)}(β−1) = ∑_{m≥0} i(F_{G̃}, m) β^m (1−β)^{dim(F_{G̃})+1}, linking reduced forms to Ehrhart theory.
• The number of lattice points in the m-th dilate of F_{G̃} equals the Kostant partition function K_{G̃}(m, 0, ..., 0, -m), allowing a new expression of the Ehrhart series in terms of Kostant partition functions.
• The identity Q_G^{S(β)}(β−1) = ∑_{m≥0} K_{G̃}(m, 0, ..., 0, -m) β^m (1−β)^{#E(G)+#V(G)} generalizes [6, Theorem 3.10] to arbitrary graphs G.