Skip to main content
SpringerLink
Log in

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

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

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.

References

  1. Takahide Adachi, Osamu Iyama, and Idun Reiten, \(\tau \)-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452.

  2. Takuma Aihara, Tilting-connected symmetric algebras, Algebr. Represent. Theory 16 (2013), no. 3, 873–894.

    Article  MathSciNet  MATH  Google Scholar 

  3. Salah Al-Nofayee, Simple objects in the heart of a\(t\)-structure, J. Pure Appl. Algebra 213 (2009), no. 1, 54–59.

    Article  MathSciNet  MATH  Google Scholar 

  4. Drew Armstrong, Christian Stump, and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4121–4151. MR 3055691

    Article  MathSciNet  MATH  Google Scholar 

  5. Sota Asai, Semibricks, Int. Math. Res. Not. IMRN 2020 (2020), no. 16, 4993–5054.

    Article  MathSciNet  MATH  Google Scholar 

  6. Sota Asai, The wall-chamber structures of the real Grothendieck groups, Adv. Math. 381 (2021).

  7. Sota Asai, Bricks over preprojective algebras and join-irreducible elements in Coxeter groups, J. Pure Appl. Algebra 226 (2022).

  8. Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras, Cambridge University Press, Cambridge, 2006.

    Book  MATH  Google Scholar 

  9. Emily Barnard, Andrew T. Carroll, and Shijie Zhu, Minimal inclusions of torsion classes, Algebraic Combin. 2 (2019), no. 5, 879–901.

    Article  MathSciNet  MATH  Google Scholar 

  10. Emily Barnard, Gordana Todorov, and Shijie Zhu, Dynamical combinatorics and torsion classes, J. Pure Appl. Algebra 225 (2021), no. 9.

  11. Thomas Brüstle and Dong Yang, Ordered exchange graphs, Advances in Representation Theory of Algebras (David J. Benson, Hennig Krause, and Andrzej Skowroński, eds.), EMS Series of Congress Reports, vol. 9, European Mathematical Society, 2013.

  12. Thomas Brüstle, David Smith, and Hipolito Treffinger, Wall and chamber structure for finite-dimensional algebras, Adv. Math. 354 (2019).

  13. Thomas Brüstle, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, and Emine Yıldırım, On the combinatorics of gentle algebras, Canad. J. Math. 72 (2020), 1551–1580.

    Article  MathSciNet  MATH  Google Scholar 

  14. Aslak Bakke Buan and Bethany R. Marsh, A category of wide subcategories, Int. Math. Res. Not. IMRN rnz082 (2019).

  15. M. C. R. Butler and C. M. Ringel, Auslander-reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179.

    Article  MathSciNet  MATH  Google Scholar 

  16. W. W. Crawley-Boevey, Maps between representations of zero-relation algebras, J. Algebra 126 (1989), no. 2, 259–263.

    Article  MathSciNet  MATH  Google Scholar 

  17. Laurent Demonet, Osamu Iyama, and Gustavo Jasso, \(\tau \)-tilting finite algebras, bricks, and\(g\)-vectors, Int. Math. Res. Not. IMRN 2019 (2019), no. 3, 852–892.

    Article  MathSciNet  MATH  Google Scholar 

  18. Laurent Demonet, Osamu Iyama, Nathan Reading, Idun Reiten, and Hugh Thomas, Lattice theory of torsion classes, arXiv:1711.01785.

  19. Changjian Fu, \(c\)-vectors via\(\tau \)-tilting theory, J. Algebra 473 (2017), 194–220.

    Article  MathSciNet  MATH  Google Scholar 

  20. Alexander Garver and Thomas McConville, Lattice propertiess of oriented exchange graphs and torsion classes, Algebr. Represent. Theory 22 (2019), no. 1, 43–78.

    Article  MathSciNet  MATH  Google Scholar 

  21. Alexander Garver and Thomas McConville, Oriented flip graphs, noncrossing tree partitions, and representation theory of tiling algebras, Glasg. Math. J. 62 (2020), no. 1, 147–182.

    Article  MathSciNet  MATH  Google Scholar 

  22. Eric J. Hanson and Kiyoshi Igusa, Pairwise compatibility for 2-simple minded collections, J. Pure Appl. Algebra 225 (2021), no. 6.

  23. Eric J. Hanson and Kiyoshi Igusa, \(\tau \)-cluster morphism categories and picture groups, Comm. Algebra 49 (2021), no. 10, 4376–4415.

    Article  MathSciNet  MATH  Google Scholar 

  24. Sam Hopkins, The CDE property for skew vexillary permutations, J. Combin. Theory Ser. A 168 (2019), 164–218. MR 3968125

    Article  MathSciNet  MATH  Google Scholar 

  25. Kiyoshi Igusa, The category of noncrossing partitions, arXiv:1411.0196.

  26. Kiyoshi Igusa and Gordana Todorov, Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041.

  27. Kiyoshi Igusa and Gordana Todorov, Picture groups and maximal green sequences, Electron. Res. Arc. (2021), no. 1935-9179_2021025.

  28. Kiyoshi Igusa, Gordana Todorov, and Jerzy Weyman, Picture groups of finite type and cohomology in type\({A}_n\), arXiv:1609.02636.

  29. Kiyoshi Igusa, Kent Orr, Gordana Todorov, and Jerzy Weyman, Cluster complexes via semi-invariants, Compos. Math. 145 (2009), no. 4, 1001–1034.

    Article  MathSciNet  MATH  Google Scholar 

  30. Colin Ingalls and Hugh Thomas, Noncrossing partitions and representations of quivers, Compos. Math, 145 (2009), no. 6, 1533–1562.

    Article  MathSciNet  MATH  Google Scholar 

  31. Osamu Iyama, Idun Reiten, Hugh Thomas, and Gordana Todorov, Lattice structure of torsion classes for path algebras, B. Lond. Math. Soc. 47 (2015), no. 4, 639–650.

    Article  MathSciNet  MATH  Google Scholar 

  32. Osamu Iyama, Nathan Reading, Idun Reiten, and Hugh Thomas, Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math. 154 (2018), no. 6, 1269–1305.

    Article  MathSciNet  MATH  Google Scholar 

  33. Gustavo Jasso, Reduction of\(\tau \)-tilting modules and torsion pairs, Int. Math. Res. Not. IMRN 2015 (2014), no. 16, 7190–7237.

    Article  MathSciNet  MATH  Google Scholar 

  34. Haibo Jin, Reductions of triangulated categoriess and simple minded collections, arXiv:1907.05114.

  35. Bernhard Keller and Laurent Demonet, A survey on maximal green sequences, Representation Theory and Beyond (J. Šťovíček and J. Trlifaj, eds.), Contemp. Math., vol. 758, Amer. Math. Soc., Providence RI, 2020, pp. 267–286.

  36. A. D. King, Moduli of representations of finite dimensional algebras, QJ Math 45 (1994), no. 4, 515–530.

    Article  MathSciNet  MATH  Google Scholar 

  37. Mark Kleiner, Approximations and almost split sequences in homologically finite subcategories, J. Algebra 198 (1997), no. 1, 135–163.

    Article  MathSciNet  MATH  Google Scholar 

  38. Steffen Koenig and Dong Yang, Silting objects, simple-minded collections,\(t\)-structures and co-\(t\)-structures for finite-dimensional algebras, Documenta Math. 19 (2014), 403–438.

    MathSciNet  MATH  Google Scholar 

  39. Yuya Mizuno, Arc diagrams and 2-term simple-minded collections of preprojective algebras of type\(A\), arXiv:2010.04353.

  40. Yuya Mizuno, Classifying\(\tau \)-tilting modules over preprojective algebras of Dynkin type, Math. Z. 277 (2014), no. 3-4, 665–690.

    Article  MathSciNet  MATH  Google Scholar 

  41. Nathan Reading, Noncrosssing arc diagramss and canonical join representations, SIAM J. Discrete Math. 29 (2015), no. 2, 736–750.

    Article  MathSciNet  MATH  Google Scholar 

  42. Nathan Reading, David Speyer, and Hugh Thomas, The fundamental theorem of finite semidistributive lattices, Selecta Math. 27 (2021), no. 59.

  43. C. M. Ringel, Representations of k-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302.

    Article  MathSciNet  MATH  Google Scholar 

  44. Claus Michael Ringel, The Catalan combinatorics of the hereditary Artin algebras, Recent developments in representation theory, Contemp. Math., vol. 673, Amer. Math. Soc., Providence, RI, 2016, pp. 51–177. MR 3546710

  45. Jan Schröer, Modules without self-extensions over gentle algebras, J. Algebra 216 (1999), no. 1, 178–189.

    Article  MathSciNet  MATH  Google Scholar 

  46. Hugh Thomas and Nathan Williams, Independence posets, J. Comb. 10 (2019), no. 3, 545–578. MR 3960513

    MathSciNet  MATH  Google Scholar 

  47. Hugh Thomas and Nathan Williams, Rowmotion in slow motion, Proc. Lond. Math. Soc. 119 (2019), no. 5, 1149–1178.

    Article  MathSciNet  MATH  Google Scholar 

  48. Hipolito Treffinger, On sign-coherence of\(c\)-vectors, J. Pure Appl. Algebra 223 (2019), no. 6, 2382–2400.

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, DePaul Universsity, Chicago, IL, 60604, USA

    Emily Barnard

  2. Department of Mathematics, Brandeis University, Waltham, MA, 02453, USA

    Eric J. Hanson

Authors
  1. Emily Barnard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric J. Hanson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric J. Hanson.

Ethics declarations

Conflicts of interest

The authors have no competing interests to declare that are relevant to the content of this article. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Additional information

Communicated by Nathan Williams.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barnard, E., Hanson, E.J. Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank. Ann. Comb. 26, 803–855 (2022). https://doi.org/10.1007/s00026-022-00585-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-022-00585-4

Mathematics Subject Classification

Search

Navigation