Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank

TL;DR

This paper characterizes the conditions for completing semibrick pairs in certain finite-dimensional algebras, constructs a combinatorial model for preprojective algebras of type A, and shows limitations of pairwise criteria in larger Dynkin diagrams.

Contribution

It provides a characterization of semibrick pair completability for small $ au$-tilting finite algebras and develops a combinatorial model for preprojective algebras of type A.

Findings

Semibrick pair completability characterized by pairwise conditions for small algebras

Constructed a combinatorial model using the weak order for type A preprojective algebras

No pairwise criteria exist for larger Dynkin diagrams with more than 3 vertices

Abstract

Let $\Lambda$ be a finite-dimensional associative algebra over a field. A semibrick pair is a finite set of $\Lambda$-modules for which certain Hom- and Ext-sets vanish. A semibrick pair is completable if it can be enlarged so that a generating condition is satisfied. We prove that if $\Lambda$ is $\tau$-tilting finite with at most 3 simple modules, then the completability of a semibrick pair can be characterized using conditions on pairs of modules. We then use the weak order to construct a combinatorial model for the semibrick pairs of preprojective algebras of type $A_n$. From this model, we deduce that any semibrick pair of size $n$ satisfies the generating condition, and that the dimension vectors of any semibrick pair form a subset of the column vectors of some $c$-matrix. Finally, we show that no "pairwise" criteria for completability exists for preprojective algebras of Dynkin diagrams with more than 3 vertices.