A $\tau$-Tilting Approach to Dissections of Polygons

TL;DR

This paper establishes a connection between polygon dissections and support τ-tilting theory by showing that accordion complexes are isomorphic to support τ-tilting complexes of certain finite-dimensional algebras.

Contribution

It introduces a novel correspondence between polygon dissections and support τ-tilting complexes, expanding the understanding of their combinatorial and algebraic structures.

Findings

Accordion complexes are isomorphic to support τ-tilting complexes of explicit finite-dimensional algebras.

Proves properties of induced subcomplexes of support τ-tilting complexes.

Provides a new algebraic perspective on polygon dissections.

Abstract

We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $\tau$-tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced subcomplexes of support $\tau$-tilting simplicial complexes of finite dimensional algebras.