Flow cones of graphs with cycles and locally gentle algebras

TL;DR

The work builds a bridge between flow geometry on directed graphs and representation-theoretic g-vector fans for gentle and locally gentle algebras. It proves a linear isomorphism between reduced DKK fans and g-vector fans, extends the DKK framework to graphs with cycles via cyclic ample framings, and unveils polytopal realizations such as the cyclohedron and a new doppelgänghedron with the same f-vector as a permutahedron. It also develops cyclic volume flows that realize the flow polytope volumes and extends Harder–Narasimhan polytope theory to locally gentle quivers, connecting shard polytopes and arc combinatorics. Together, these results yield new algorithmic and geometric tools for enumerating maximal cones, volumes, and polyhedral realizations in both combinatorial and algebraic settings, while linking known polytopes to broader fan structures via Minkowski sums and projections.

Abstract

Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a $g$-vector fan of a gentle quiver associated to the graph was discovered by von Bell, Braun, Bruegge, Hanely, Peterson, Serhiyenko, and Yip in 2022. This correspondence has been fruitful in uncovering lattice structures in the triangulations. We start by showing that this correspondence is actually a linear isomorphism. We then consider flow cones of certain graphs with cycles. For this case, we give a DKK-like triangulation of the cone, and extend the correspondence to the finite $g$-vector fan of a corresponding locally gentle quiver. In addition, we extend to cyclic graphs a mysterious result of Postnikov--Stanley and Baldoni--Vergne, giving the volume of flow polytopes of acyclic graphs as the number of certain integer flows on the same graph. We illustrate our results with a two-parameter family of cyclic graphs that includes a cycle graph and nested 2-cycles as special cases. We show that the fans of its DKK-like triangulations are respectively isomorphic to the normal fan of the cyclohedron and of a new polytope with the same $f$-vector but different combinatorial type than the permutahedron.

Figures (21)

• Figure 1: (A) A full DAG $G = {\raisebox{-.8mm}{$\bm{\diagup}\bm{\diagdown}$}\raisebox{.8mm}{$\bm{\diagup}\bm{\diagdown}$}}$ with an ample framing $F$ (at each vertex, $\text{red} \prec \text{blue}$), (B) the routes of $(G,F)$ of which the last three are exceptional, and (C) illustration of the dual of the triangulation ${\sf DKK}_{\rm red}(G,F)$.
• Figure 2: The five volume integer flows from \ref{['subfig:full_graph_and_routes3']} (only edges with non-zero flows are labelled). Below each flow, we list the clique that corresponds to it via the bijection $\Phi^{-1}$. We do not list the exceptional routes $E_1,E_2,E_3$, which are present in each clique.
• Figure 3: The $g$-vector fan $\pmb{g}{\sf fan}(Q)$ of the quiver $Q$ in \ref{['ex:g-vector_running_example']}. The vector $\pmb{g}_{M}$ is simply labelled by $M$.
• Figure 4: The blossoming quiver $(Q^\text{❀},R^\text{❀})$ for the gentle quiver $(Q,R)$ from \ref{['ex:runningGQR']}, with the maximal string (drawn with thick arrows) corresponding to the lazy path at $1$.
• Figure 5: Illustration of a substring $\sigma$ at the top of $\rho$ (left), and a substring $\sigma'$ at the bottom of $\rho$ (right).

Theorems & Definitions (172)

• Theorem 1.1: \ref{['prop:linear-iso']}
• Theorem 1.2: \ref{['thm:connection_g-vector_NKred']}
• Proposition 1.3: \ref{['prop:locallygentlegvectorfan']}
• Theorem 1.4: \ref{['cor:DKKred-is-polytopal']}
• Theorem 1.5: \ref{['thm:cyclic-DKK']}
• Theorem 1.6: \ref{['thm:normVolCyclicIntegerFlow']}
• Theorem 1.7: \ref{['thm:hkr-vol']}
• Theorem 1.8: \ref{['thm:f-vect_doppel']}
• Remark 2.2
• Definition 2.3: DKK