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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.

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Notes

  1. The word compatibility is used in place of completability in [22]. We have chosen to use the term completability since, a priori, determining whether a semibrick pair is completable is not characterized internally.

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Acknowledgements

The authors are thankful to Kiyoshi Igusa, Haibo Jin, Job Rock, Hugh Thomas, Gordana Todorov, and John Wilmes for insightful discussions and support. A large portion of this work is included in EH’s Ph.D thesis, and a portion of this work was completed while EH was affiliated with the Norwegian University of Science and Technology (NTNU). EH thanks NTNU for their support and hospitality. The authors are also thankful to a pair of anonymous referees for their suggestions on how to improve this paper.

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Barnard, E., Hanson, E.J. Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00585-4

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Mathematics Subject Classification

  • 16G20
  • 05E10