Geometric Realizations of the Accordion Complex of a Dissection

  • Thibault Manneville
  • Vincent Pilaud


Consider 2n points on the unit circle and a reference dissection \({\mathrm {D}}_\circ \) of the convex hull of the odd points. The accordion complex of \({\mathrm {D}}_\circ \) is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of \({\mathrm {D}}_\circ \). In particular, this complex is an associahedron when \({\mathrm {D}}_\circ \) is a triangulation and a Stokes complex when \({\mathrm {D}}_\circ \) is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection \({\mathrm {D}}_\circ \), generalizing known constructions arising from cluster algebras.


Permutahedra Zonotopes Associahedra g-, c- and d-Vectors 

Mathematics Subject Classification

52B11 52B12 13F60 



We thank C. Hohlweg and S. Stella for many helpful discussions on realizations of the associahedron [27] which were the starting point of this paper. We are grateful to F. Chapoton for various conversations on quadrangulations and Stokes posets, and to A. Garver and T. McConville for introducing us with the accordion complexes during FPSAC’16. Their works [9, 20] were highly inspiring and motivating. We also thank N. Thiery for a question which led to the approach of Sect. 5.2, and to P.-G. Plamondon for discussions on the generalization to cluster algebras presented in Sect. 5.3. Finally, we are grateful to two anonymous referees for their attentive reading and their suggestions on the content and presentation which largely improved our original draft.


  1. 1.
    Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-Tilting theory. Compos. Math. 150(3), 415–452 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baryshnikov, Y.: On stokes sets. In: Siersma, D., et al. (eds.) New Developments in Singularity Theory (Cambridge, 2000). NATO Science Series II: Mathematics, Physics and Chemistry, vol. 21, pp. 65–86. Kluwer, Dordrecht (2001)Google Scholar
  3. 3.
    Bateni, A.H., Manneville, T., Pilaud, V.: A note on quadrangulations and Stokes complexes (2016). In preparationGoogle Scholar
  4. 4.
    Billera, L.J., Filliman, P., Sturmfels, B.: Constructions and complexity of secondary polytopes. Adv. Math. 83(2), 155–179 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brüstle, T., Douville, G., Mousavand, K., Thomas, H., Yıldırım, E.: On the combinatorics of gentle algebras (2017). arXiv:1707.07665
  6. 6.
    Brüstle, T., Dupont, G., Pérotin, M.: On maximal green sequences. Int. Math. Res. Not. IMRN 2014(16), 4547–4586 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carr, M.P., Devadoss, S.L.: Coxeter complexes and graph-associahedra. Topol. Appl. 153(12), 2155–2168 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceballos, C., Santos, F., Ziegler, G.M.: Many non-equivalent realizations of the associahedron. Combinatorica 35(5), 513–551 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chapoton, F.: Stokes posets and serpent nests. Discret. Math. Theor. Comput. Sci. 18(3), Art. No. 18 (2016)Google Scholar
  10. 10.
    Chapoton, F., Fomin, S., Zelevinsky, A.: Polytopal realizations of generalized associahedra. Can. Math. Bull. 45(4), 537–566 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)Google Scholar
  12. 12.
    Demonet, L., Iyama, O., Jasso, G.: \(\tau \)-Tilting finite algebras, bricks, and g-vectors. Int. Math. Res. Not. IMRN.
  13. 13.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. 62(4), 437–468 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fomin, S., Thurston, D.: Cluster algebras and triangulated surfaces. Part II: Lambda lengths (2012). arXiv:1210.5569
  16. 16.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fomin, S., Zelevinsky, A.: \(Y\)-systems and generalized associahedra. Ann. Math. 158(3), 977–1018 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garver, A., McConville, T.: Oriented flip graphs and noncrossing tree partitions (2016). arXiv:1604.06009
  21. 21.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Modern Birkhäuser Classics. Birkhäuser, Boston (2008). Reprint of the 1994 editionGoogle Scholar
  22. 22.
    Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Am. Math. Soc. 31(2), 497–608 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Haiman, M.: Constructing the associahedron (1984).
  24. 24.
    Hohlweg, C.: Permutahedra and associahedra. In: Müller-Hoissen, F., Pallo, J.M., Stasheff, J. (eds.): Associahedra, Tamari Lattices and Related Structures. Tamari Memorial Festschrift. Progress in Mathematical Physics, vol. 299. Springer, Basel (2012), pp. 129–159Google Scholar
  25. 25.
    Hohlweg, C., Lange, C.E.M.C.: Realizations of the associahedron and cyclohedron. Discret. Comput. Geom. 37(4), 517–543 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hohlweg, C., Lange, C.E.M.C., Thomas, H.: Permutahedra and generalized associahedra. Adv. Math. 226(1), 608–640 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hohlweg, C., Pilaud, V., Stella, S.: Polytopal realizations of finite type \(\mathbf{g}\)-vector fans. Adv. Math. 328, 713–749 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lee, C.W.: The associahedron and triangulations of the \(n\)-gon. Eur. J. Comb. 10(6), 551–560 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Loday, J.-L.: Realization of the Stasheff polytope. Arch. Math. 83(3), 267–278 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Manneville, T., Pilaud, V.: Compatibility fans for graphical nested complexes. J. Comb. Theory Ser. A 150, 36–107 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Müller-Hoissen, F., Pallo, J.M., Stasheff, J. (eds.): Associahedra, Tamari Lattices and Related Structures. Tamari Memorial Festschrift. Progress in Mathematical Physics, vol. 299. Springer, Basel (2012)Google Scholar
  32. 32.
    Palu, Y., Pilaud, V., Plamondon, P.-G.: Non-kissing complexes and \(\tau \)-tilting for gentle algebras (2017). arXiv:1707.07574
  33. 33.
    Pilaud, V.: Signed tree associahedra (2013). arXiv:1309.5222
  34. 34.
    Pilaud, V., Plamondon, P.-G., Stella, S.: A \(\tau \)-tilting approach to dissections of polygons (2017). arXiv:1710.02119
  35. 35.
    Pilaud, V., Santos, F.: The brick polytope of a sorting network. Eur. J. Comb. 33(4), 632–662 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pilaud, V., Stump, C.: Brick polytopes of spherical subword complexes and generalized associahedra. Adv. Math. 276, 1–61 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Postnikov, A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN 2009(6), 1026–1106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Reading, N.: Cambrian lattices. Adv. Math. 205(2), 313–353 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Reading, N.: Sortable elements and Cambrian lattices. Algebra Univers. 56(3–4), 411–437 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Reading, N., Speyer, D.E.: Cambrian fans. J. Eur. Math. Soc. 11(2), 407–447 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Shnider, S., Sternberg, S.: Quantum Groups: From Coalgebras to Drinfeld Algebras. Graduate Texts in Mathematical Physics, vol. 2. International Press, Cambridge (1993)Google Scholar
  42. 42.
    Stasheff, J.: Homotopy associativity of \(H\)-spaces I, II. Trans. Am. Math. Soc. 108(2), 293–312 (1963)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Stella, S.: Polyhedral models for generalized associahedra via Coxeter elements. J. Algebr. Comb. 38(1), 121–158 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tamari, D.: Monoides préordonnés et chaînes de Malcev. Ph.D. thesis, Université Paris Sorbonne (1951)Google Scholar
  45. 45.
    Zelevinsky, A.: Nested complexes and their polyhedral realizations. Pure Appl. Math. Q. 2(3), 655–671 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ziegler, G.M.: Lectures on Polytopes. Graduate texts in Mathematics, vol. 152. Springer, New York (1995)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LIX, École PolytechniquePalaiseauFrance
  2. 2.CNRS & LIX, École PolytechniquePalaiseauFrance

Personalised recommendations