Non-kissing complexes and tau-tilting for gentle algebras

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.

72 pages, 45 figures; Version 4: Minor corrections (in particular Fig. 19), published version

References in corpus (7)

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Cited by in corpus (12)

• Lattice theory of torsion classes: Beyond $\tau$-tilting theory
• Geometric realizations of the accordion complex of a dissection
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Celebrating Loday's Associahedron
• On extensions for gentle algebras
• A $\tau$-Tilting Approach to Dissections of Polygons
• On $\tau$-tilting modules over trivial extensions of gentle tree algebras
• A multiplication formula for cluster characters in gentle algebras
• Wigglyhedra
• On homomorphisms and generically $\tau$-regular components for skewed-gentle algebras
• Jordan recoverability of some subcategories of modules over gentle algebras
• Classifying torsion classes of gentle algebras