Semistable subcategories for tiling algebras

TL;DR

This paper establishes a correspondence between semistable subcategories of tiling algebras and noncrossing tree partitions, extending known results from Dynkin type A to a broader class of algebras.

Contribution

It introduces a bijection between semistable subcategories of tiling algebras and noncrossing tree partitions, showing an isomorphism of their posets.

Findings

Semistable subcategories correspond to noncrossing tree partitions.

The poset of semistable subcategories is isomorphic to the poset of noncrossing tree partitions.

Generalizes the Dynkin type A case to tiling algebras.

Abstract

Semistable subcategories were introduced in the context of Mumford's GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding poset of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Coelho Sim{\~o}es and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Moreover, this bijection defines an isomorphism of the posets on these objects. Our work recovers that of Ingalls and Thomas in Dynkin type $A$.