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Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

Matias von Bell, Benjamin Braun, Kaitlin Bruegge, Derek Hanely, Zachery Peterson, Khrystyna Serhiyenko, Martha Yip

Abstract

The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and $τ$-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart $h^\ast$-polynomials.

Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

Abstract

The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and -tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart -polynomials.
Paper Structure (12 sections, 41 theorems, 29 equations, 18 figures)

This paper contains 12 sections, 41 theorems, 29 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Background on Flow Polytopes
  3. Flows and Flow Polytopes
  4. Framings, Coherent Routes, and the DKK Triangulation
  5. Ample Framings
  6. Full Framed Graphs
  7. Valid Framed Graphs
  8. Enumerating Ample Framings
  9. Bijection between $KQ/I$-modules and Non-exceptional Routes
  10. Background on Path Algebras and Gentle Algebras
  11. The Bijection
  12. Gorenstein Flow Polytopes

Key Result

Proposition 2.3

Given a DAG $G$, the set of flows $\mathcal{F}(G)$ forms a vector subspace of $\mathbb{R}^{|E|}$ spanned by the characteristic vectors of the routes and has dimension The vertices of $\mathcal{F}_1$ are the characteristic vectors $\{v_R : R\in \mathcal{P}(G)\}$ and

Figures (18)

  • Figure 1: The graph $\mathop{\mathrm{car}}\nolimits(8)$.
  • Figure 2: Example of a complete contraction of the graph $\mathop{\mathrm{car}}\nolimits(8)$ where the first and last idle edges have been contracted. The left graph shows a framing at the inner vertices labeled in orange. This framing induces a labeling on each edge $(u,v)$ representing its ordering in both $\mathrm{in}(v)$ and $\mathrm{out}(u)$. The induced edge-labeling is shown in blue on the right.
  • Figure 3: Given the length framing of the complete contraction of the graph $\mathop{\mathrm{car}}\nolimits(8)$ described in Example \ref{['ex:car8framing']}, the above routes form a maximal clique. The routes in the left column are the exceptional routes for this framing.
  • Figure 4: The adjacency graph from Example \ref{['ex:adjacencygraph']} with vertices consisting of the four non-exceptional routes listed in the right-hand column of Figure \ref{['fig:exceptional routes of contracted car(8)']}.
  • Figure 5: The graph $G$ from Example \ref{['ex:disjointpathscycles']}.
  • ...and 13 more figures

Theorems & Definitions (113)

  • Example 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 103 more