\(h\)-polynomials via reduced forms

Karola Meszaros¹

¹Cornell University

The electronic journal of combinatorics, Tome 22 (2015) no. 4

Résumé

The flow polytope $F_{\widetilde{G}}$ 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 $F_{\widetilde{G}}$ 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.

Détail

DOI : 10.37236/5172

Classification : 05E45, 05C21, 52B20
Mots-clés : flow polytope, triangulation, \(h\)-polynomial, nonnegativity, reduced form, subdivision algebra