Overview
This research introduces and characterizes the 'turbulence polyhedron' for arbitrary gentle algebras, building upon established connections between unit flows on directed acyclic graphs (DAGs) and the representation theory of certain gentle algebras. The study demonstrates that support $\tau$-tilting modules of a gentle algebra index a specific unimodular triangulation of its associated turbulence polyhedron. Furthermore, for gentle algebras exhibiting representation-infinite characteristics, the research details two distinct larger polyhedral dissections that incorporate lower-dimensional walls.
The work establishes a quotient map from the turbulence polyhedron to a newly defined 'g-polyhedron', situated within the ambient space of the g-vector fan. This mapping serves to prove that gentle algebras are g-convex. Additionally, the images of the identified wall types under this quotient map provide two different interpretations for the complement of the g-vector fan specific to gentle algebras.
Research Context
The space of unit flows on a directed acyclic graph (DAG) has previously been recognized for its capacity to admit regular unimodular triangulations. These triangulations are known to be induced by specific framings of the DAG. A recent development in this field has connected amply framed DAGs and their corresponding triangulated flow polytopes to the representation theory of certain gentle algebras. This earlier work established a relationship between these geometric structures and algebraic concepts within representation theory.
Approach
The methodology involved extending the aforementioned connection by defining a flow directly on the fringed quiver of an arbitrary gentle algebra. This novel construction led to the definition of the 'turbulence polyhedron' as the space of these unit flows. The subsequent analytical steps focused on examining the properties of this newly defined polyhedron and its relationship with the algebraic structures of the gentle algebra.
The study then investigated the support $\tau$-tilting modules of a gentle algebra. It was shown that these modules serve to index a unimodular triangulation of the turbulence polyhedron. For cases where the gentle algebra was of representation-infinite type, the research further explored and described two distinct larger polyhedral dissections. These dissections were formed by augmenting the existing picture with the inclusion of lower-dimensional walls.
A key part of the approach involved defining a 'g-polyhedron' and establishing a quotient map from the turbulence polyhedron to this g-polyhedron. This g-polyhedron is positioned within the ambient space of the g-vector fan. The images of the two types of walls identified in the representation-infinite case, when mapped under this quotient, were then analyzed to provide interpretations for the complement of the g-vector fan of a gentle algebra.
Findings
- A 'turbulence polyhedron' was defined as the space of unit flows on the fringed quiver of an arbitrary gentle algebra.
- Support $\tau$-tilting modules of a gentle algebra index a unimodular triangulation of its turbulence polyhedron.
- In the representation-infinite case, this triangulation is incomplete.
- Two distinct larger polyhedral dissections are possible in the representation-infinite case, achieved by adding lower-dimensional walls.
- The turbulence polyhedron has a quotient map to a 'g-polyhedron', located in the ambient space of the g-vector fan.
- This quotient map demonstrates that gentle algebras are g-convex.
- The images of the two types of walls under the quotient map provide two different interpretations for the complement of the g-vector fan of a gentle algebra.