Flows on Gentle Algebras

TL;DR

This paper explores the geometric and combinatorial structures associated with gentle algebras, connecting flow polytopes, triangulations, and representation theory, and introduces new polyhedral dissections and convexity properties.

Contribution

It defines the turbulence polyhedron for gentle algebras, links support tau-tilting modules to its triangulation, and establishes g-convexity and new polyhedral interpretations.

Findings

Support tau-tilting modules index a unimodular triangulation.

Infinite case triangulation is extended with additional polyhedral dissections.

The turbulence polyhedron maps onto the g-polyhedron, demonstrating g-convexity.

Abstract

The space of unit flows on a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. Amply framed DAGs and their triangulated flow polytopes have recently been connected with the representation theory of certain gentle algebras. We expand on this connection by defining a flow on the fringed quiver of an arbitrary gentle algebra. We call the space of unit flows its turbulence polyhedron. We show that support tau-tilting modules of a gentle algebra index a unimodular triangulation of its turbulence polyhedron. In the representation-infinite case, this triangulation is not complete and we give two different larger polyhedral dissections given by adding lower-dimensional walls to the picture. The turbulence polyhedron has a quotient map to what we define as the g-polyhedron lying in the ambient space of the g-vector fan, proving that gentle algebras are g-convex. Moreover, the images of our two types of walls under this quotient map provide two different interpretations for the complement of the g-vector fan of a gentle algebra.