Skip to main content
SpringerLink
Log in
Book cover

Associahedra, Tamari Lattices and Related Structures pp 293–322Cite as

Birkhäuser

From the Tamari Lattice to Cambrian Lattices and Beyond

  • Chapter
  • First Online:

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

Abstract

We trace the path from the Tamari lattice, via lattice congruences of the weak order, to the definition of Cambrian lattices in the context of finite Coxeter groups, and onward to the construction of Cambrian fans. We then present sortable elements, the key combinatorial tool for studying Cambrian lattices and fans. The chapter concludes with a brief description of the applications of Cambrian lattices and sortable elements to Coxeter-Catalan combinatorics and to cluster algebras.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Amiot, “A derived equivalence between cluster equivalent algebras”, J. Algebra 351 (2012) 107–129.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Amiot, O. Iyama, I. Reiten and G. Todorov, “Preprojective algebras and c-sortable words”, Proc. London Math. Soc. (3) 104 (2012) 513–539.

    Google Scholar 

  3. D. Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc., vol. 202, no. 949, Amer. Math. Soc., Providence, RI, 2009.

    Google Scholar 

  4. C.A. Athanasiadis, T. Brady, J. McCammond and C. Watt, “h-vectors of generalized associahedra and noncrossing partitions”, Int. Math. Res. Not. 2006, Art. ID 69705, 28 pp.

    Google Scholar 

  5. L. Billera and B. Sturmfels, “Fiber polytopes”, Ann. of Math. (2) 135 (1992) 527–549.

    Google Scholar 

  6. L. Billera and B. Sturmfels, “Iterated Fiber polytopes”, Mathematika 41 (1994) 348–363.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Björner and M. Wachs, “Shellable nonpure complexes and posets. II.”, Trans. Amer. Math. Soc. 349 (1997) 3945–3975.

    Google Scholar 

  8. T. Brady and C. Watt, “Non-crossing partition lattices in finite real reflection groups”, Trans. Amer. Math. Soc. 360 (2008) 1983–2005.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Chajda and V. Snášel, “Congruences in Ordered Sets”, Math. Bohem. 123 (1998) 95–100.

    Google Scholar 

  10. F. Chapoton, S. Fomin and A. Zelevinsky, “Polytopal realizations of generalized associahedra”, Canad. Math. Bull. 45 (2002) 537–566.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Dorfer, “Lattice-extensions by means of convex sublattices”, in Contributions to general algebra (Linz, 1994), vol. 9, G. Pilz, ed., H¨older-Pichler-Tempsky, Vienna, 1995, 127–132.

    Google Scholar 

  12. S. Fomin and N. Reading, “Root systems and generalized associahedra”, in Geometric Combinatorics, E. Miller, V. Reiner and B. Sturmfels, eds., IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, 63–131.

    Google Scholar 

  13. S. Fomin and A. Zelevinsky, “Y-systems and generalized associahedra”, Ann. of Math. 158 (2003) 977–1018.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Fomin and A. Zelevinsky, “Cluster algebras. I. Foundations”, J. Amer. Math. Soc. 15 (2002) 497–529.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Fomin and A. Zelevinsky, “Cluster Algebras II: Finite Type Classification”, Inventiones Mathematicae 154 (2003) 63–121.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Fomin and A. Zelevinsky, “Cluster algebras: notes for the CDM-03 conference”, in Current Developments in Mathematics, A.J. de Jong, D. Jerison, G. Lustig, B. Mazur, W. Schmid and S.T. Yau, eds., Int. Press, Somerville, MA, 2003, 1–34.

    Google Scholar 

  17. S. Fomin and A. Zelevinsky, “Cluster Algebras IV: Coefficients”, Compositio Mathematica 143 (2007) 112–164.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Forcey, “Extending the Tamari lattice to some compositions of species”, in this volume.

    Google Scholar 

  19. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh and J.-Y. Thibon, “Noncommutative symmetric functions”, Adv. Math. 112 (1995) 218–348.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Giraudo, “Algebraic and combinatorial structures on Baxter permutations”, arxiv.org/ abs/1011.4288. .

  21. C. Hohlweg, “Generalized associahedra from the geometry of finite reflection groups”, in this volume.

    Google Scholar 

  22. C. Hohlweg, C. Lange and H. Thomas, “Permutahedra and generalized associahedra”, Adv. Math. 226 (2011) 608–640.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. Ingalls and H. Thomas, “Noncrossing partitions and representations of quivers”, Compositio Mathematica 145 (2009) 1533–1562.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Kleiner and A. Pelley, “Admissible sequences, preprojective representations of quivers, and reduced words in the Weyl group of a Kac-Moody algebra”, Int. Math. Res. Not. 2007, no. 4, Art. ID rnm013, 28 pp.

    Google Scholar 

  25. M. Kolibiar, “Congruence relations and direct decompositions of ordered sets”, Acta Sci. Math. (Szeged) 51 (1987) 129–135.

    MathSciNet  Google Scholar 

  26. G. Kreweras, “Sur les partitions non croisées d’un cycle”, Discrete Math. 1 (1972) 333–350.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Law, “The Hopf algebra of sashes”, in preparation (2012).

    Google Scholar 

  28. S. Law and N. Reading, “The Hopf algebra of diagonal rectangulations”, J. Combin. Theory Ser. A 119 (2012) 788–824.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.-L. Loday and M. Ronco, “Hopf algebra of planar binary trees”, Adv. Math. 139 (1998) 293–309.

    Article  MathSciNet  MATH  Google Scholar 

  30. J.-L. Loday and M. Ronco, “Order structure on the algebra of permutations and of planar binary trees”, J. Algebraic Combin. 15 (2002) 253–270.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Rambau and V. Reiner, “A survey of the higher Stasheff-Tamari orders”, in this volume.

    Google Scholar 

  32. N. Reading, “Order Dimension, Strong Bruhat Order and Lattice Properties for Posets”, Order 19 (2002) 73–100.

    Article  MathSciNet  MATH  Google Scholar 

  33. N. Reading, “Lattice and Order Properties of the Poset of Regions in a Hyperplane Arrangement”, Algebra Universalis 50 (2003) 179–205.

    Article  MathSciNet  MATH  Google Scholar 

  34. N. Reading, “Lattice congruences of the weak order”, Order 21 (2004) 315–344.

    Article  MathSciNet  MATH  Google Scholar 

  35. N. Reading, “Lattice congruences, fans and Hopf algebras”, J. Combin. Theory Ser. A 110 (2005) 237–273.

    Google Scholar 

  36. N. Reading, “Cambrian Lattices”, Adv. Math. 205 (2006) 313–353.

    Article  MathSciNet  MATH  Google Scholar 

  37. N. Reading, “Clusters, Coxeter-sortable elements and noncrossing partitions”, Trans. Amer. Math. Soc. 359 (2007) 5931–5958.

    Article  MathSciNet  MATH  Google Scholar 

  38. N. Reading, “Sortable elements and Cambrian lattices”, Algebra Universalis 56 (2007) 411–437.

    Article  MathSciNet  MATH  Google Scholar 

  39. N. Reading, “Noncrossing partitions and the shard intersection order”, J. Algebraic Combin. 33 (2011) 483–530.

    Article  MathSciNet  MATH  Google Scholar 

  40. N. Reading, “Generic rectangulations”, European J. Combin. 33 (2012) 610–623.

    Article  MathSciNet  MATH  Google Scholar 

  41. N. Reading and D. Speyer, “Cambrian Fans”, J. Eur. Math. Soc. 11 (2009) 407–447.

    Article  MathSciNet  MATH  Google Scholar 

  42. N. Reading and D. Speyer, “Sortable elements in infinite Coxeter groups”, Trans. Amer. Math. Soc. 363 (2011) 699–761.

    Article  MathSciNet  MATH  Google Scholar 

  43. N. Reading and D. Speyer, “Combinatorial frameworks for cluster algebras”, arxiv.org/abs/1111.2652. .

  44. N. Reading and D. Speyer, “Cambrian frameworks for cluster algebras of affine Cartan type”, in preparation (2012).

    Google Scholar 

  45. V. Reiner, “Equivariant fiber polytopes”, Doc. Math. 7 (2002) 113–132.

    MathSciNet  MATH  Google Scholar 

  46. L. Santocanale and F. Wehrung, “Sublattices of associahedra and permutohedra”, arxiv.org/abs/1103.3488. .

  47. R.P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  48. J.R. Stembridge, “Computational aspects of root systems, Coxeter groups, and Weyl characters”, in Interaction of combinatorics and representation theory, J.R. Stembridge, J.-Y. Thibon and M.A.A van Leeuwen, eds., MSJ Memoirs., vol. 11, Math. Soc. Japan, 2001, 1–38.

    Google Scholar 

  49. H. Thomas, “The Tamari lattice as it arises in quiver representations”, in this volume.

    Google Scholar 

  50. A. Tonks, “Relating the associahedron and the permutohedron”, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), J.-L. Loday, J.D. Stasheff and A.A. Voronov, eds., Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, 33–36.

    Google Scholar 

  51. S. Yang and A. Zelevinsky, “Cluster algebras of finite type via Coxeter elements and principal minors”, Transform. Groups 13 (2008) 855–895.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Nathan Reading

Authors
  1. Nathan Reading
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nathan Reading .

Editor information

Editors and Affiliations

  1. for Dynamics and Self-Organization, Max-Planck-Institute, Bunsenstrasse 10, Göttingen, 37073, Germany

    Folkert Müller-Hoissen

  2. , Département d'Informatique, LE2I, Université de Bourgogne, B.P. 47870, Dijon Cedex, 21078, France

    Jean Marcel Pallo

  3. , Department of Mathematics, University of North Carolina at Chapel H, CB #3250, Phillips Hall, Chapel Hill, 27599, North Carolina, USA

    Jim Stasheff

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this chapter

Cite this chapter

Reading, N. (2012). From the Tamari Lattice to Cambrian Lattices and Beyond. 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_15

Download citation

Publish with us

Policies and ethics

Search

Navigation