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 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.

v3: final version. v2: Added motivation section with applications of the pairwise 2-simple minded completability property, expanded Sections 6-7 (formerly Section 5) to emphasize the relationship with the weak order and establish a bijection between 2-colored noncrossing arc diagrams and semibrick pairs, 40 pages, 11 figures

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