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

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 three 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 three vertices.