Non-kissing complexes and tau-tilting for gentle algebras

TL;DR

This paper links support τ-tilting complexes of gentle algebras to non-kissing complexes of walks, revealing their combinatorial and geometric structures, including lattice properties and associahedra.

Contribution

It provides a new interpretation of support τ-tilting complexes as non-kissing complexes for gentle algebras, and explores their lattice and fan structures.

Findings

Support τ-tilting complex is the non-kissing complex of walks on the blossoming quiver.

The flip graph of facets forms a congruence-uniform lattice.

The g-vector fan is the normal fan of a non-kissing associahedron.

Abstract

We interpret the support $\tau$-tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its $\mathbf{g}$-vector fan and prove that it is the normal fan of a non-kissing associahedron.