Oriented Flip Graphs, Noncrossing Tree Partitions, and Representation Theory of Tiling Algebras

Abstract

The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

Semistable subcategories for tiling algebras

“…In [7], it is shown that if δ is a green or red arc in a facet of ∆ N C pT q, it gives rise to a g-vector, denoted gpδq P Z n . Additionally, in [3], it is shown that the facets of ∆ N C pT q are in bijection with wide subcategories of modpΛ T q. With these facts in mind, we arrive at our main theorem.…”

“…In the setting of tiling algebras, all strings are supported on connected acyclic subgraphs of Q T , but this is not the case in general. The string module M pwq is the representation of Q T obtained by assigning the vector space k to each vertex in the string w and identity morphisms to each arrow in w. 2 Using these facts, we obtain that the indecomposable Λ T -modules are parameterized by segments of T (i.e., acyclic paths s " pv 0 , v 1 , ..., v t q whose endpoints are interior vertices of T and any two consecutive edges pv i´1 , v i q and pv i , v i`1 q are incident to a common face) [3,Corollary 4.3]. Let wpsq denote the unique string in Λ T corresponding to the segment s P SegpT q, the set of all segments of T , and M pwpsqq the corresponding string module.…”

“…Let pv, e, F q and pu, e 1 , Gq be two green (resp., red) flags such that rv, us is a segment, a green (resp., red) admissible curve for rv, us is a simple curve σ : r0, 1s Ñ D 2 for which σp0q " zpv, F q, σp1q " zpu, Gq, and σpr0, 1sq Ď D 2 zpT zru, vs q. 3 Strictly speaking, the endpoints of σ are zpv, F q and zpu, Gq. However, for convenience, we will often refer to the endpoints of σ as simply v and u when it is not necessary to specify the exact corners where σ begins and ends.…”

“…The purpose of this work is to combinatorially classify the semistable subcategories for the class of tiling algebras, introduced in [2] to study endomorphism algebras of maximal rigid objects in some negative Calabi-Yau categories. Following [3], these algebras, denoted Λ T , are defined by the data of a tree T embedded in the disk D 2 whose interior vertices have degree at least 3 (see Figure 1). Examples of tiling algebras are given by the cluster-tilted algebras of cluster type A; the trees defining these algebras are those whose interior vertices are of degree 3.…”

The positive geometry for 𝜙p interactions

“…This generates a graph 3 which is the 1-skeleton of a convex polytope called the Accordiohedron [10,11], which we shall also call AC P p,n . 4…”