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The Tamari Lattice as it Arises in Quiver Representations

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Associahedra, Tamari Lattices and Related Structures

Part of the book series: Progress in Mathematics ((PM,volume 299))

Abstract

In this chapter, we explain how the Tamari lattice arises in the context of the representation theory of quivers, as the poset whose elements are the torsion classes of a directed path quiver, with the order relation given by inclusion.

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References

  1. C. Amiot, O. Iyama, I. Reiten, and G. Todorov, “Preprojective algebras and c-sortable words”, arxiv.org/abs/1002.4131 .

  2. I. Assem, D. Simson, and A. Skowronski, Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006.

    Google Scholar 

  3. M. Auslander, I. Reiten, and S. SmalØ, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  4. A.B. Buan and H. Krause, “Tilting and cotilting for quivers of type à n ”, J. Pure App. Algebra 190 (2004) 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Happel and L. Unger, “On a partial order of tilting modules”, Algebr. Represent. Theory 8 (2005) 147–156.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Happel and L. Unger, “On the quiver of tilting modules”, J. Algebra 284 (2005) 857–868.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Happel and L. Unger, “Reconstruction of path algebras from their posets of tilting modules”, Trans. Amer.Math. Soc. 361 (2009) 3633–3660.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Huang and D. Tamari, “Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law”, J. Combinatorial Theory Ser. A 13 (1972) 7–13.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Ingalls and H. Thomas, “Noncrossing partitions and representations of quivers”, Compos. Math. 145 (2009) 1533–1562.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Ladkani, “Universal derived equivalences of posets of cluster tilting objects”, arxiv.org/abs/0710.2860 .

  11. S. Oppermann, I. Reiten, and H. Thomas, “Quotient closed subcategories of quiver representations”, in preparation.

    Google Scholar 

  12. N. Reading, “From the Tamari lattice to Cambrian lattices and beyond”, in this volume.

    Google Scholar 

  13. C. Riedtmann and A. Schofield, “On a simplicial complex associated with tilting modules”, Comment. Math. Helv. 66 (1991) 70–78.

    Article  MathSciNet  MATH  Google Scholar 

  14. C.M. Ringel, “Minimal infinite submodule-closed subcategories”, arxiv.org/abs/1009.0864 .

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Correspondence to Hugh Thomas .

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Thomas, H. (2012). The Tamari Lattice as it Arises in Quiver Representations. In: Müller-Hoissen, F., Pallo, J., Stasheff, J. (eds) Associahedra, Tamari Lattices and Related Structures. Progress in Mathematics, vol 299. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0405-9_14

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