Cluster algebras of type \(D_4\), tropical planes, and the positive tropical Grassmannian

  • Sarah B. Brodsky
  • Cesar Ceballos
  • Jean-Philippe Labbé
Original Paper


We show that the number of combinatorial types of clusters of type \(D_4\) modulo reflection-rotation is exactly equal to the number of combinatorial types of generic tropical planes in \(\mathbb {TP}^5\). This follows from a result of Sturmfels and Speyer which classifies these generic tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian \({{\mathrm{Gr}}}(3,6)\). Speyer and Williams show that the positive part \({{\mathrm{Gr}}}^+(3,6)\) of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type \(D_4\). We provide a structural bijection between the rays of \({{\mathrm{Gr}}}^+(3,6)\) and the almost positive roots of type \(D_4\) which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type \(D_4\) to describe the equivalence of “positive” generic tropical planes in \(\mathbb {TP}^5\), giving a combinatorial model which characterizes the combinatorial types of generic tropical planes using automorphisms of pseudotriangulations of the octogon.


Tropical planes Grassmannian Pseudotriangulations Cluster complex Computational methods 

Mathematics Subject Classification

Primary 14T05 Secondary 14N10 52C30 



We are grateful to York University for hosting visits of the first and third authors. We also thank Hugh Thomas for helpful discussions.


  1. Ceballos, C., Labbé, J.-P., Stump, C.: Subword complexes, cluster complexes, and generalized multi-associahedra. J. Algebr. Combin. 39(1), 17–51 (2014)Google Scholar
  2. Ceballos, C., Pilaud, V.: Denominator vectors and compatibility degrees in cluster algebras of finite type. Trans. Am. Math. Soc. 367(2), 1421–1439 (2015a)Google Scholar
  3. Ceballos, C., Pilaud, V.: Cluster algebras of type D: pseudotriangulations approach. Electron. J. Combin. 22(4), pp. 27 (2015b)Google Scholar
  4. Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)Google Scholar
  5. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)Google Scholar
  6. Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. (2), 158(3), 977–1018 (2003a)Google Scholar
  7. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003b)Google Scholar
  8. Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)Google Scholar
  9. Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)Google Scholar
  10. Gawrilow, E., Joswig, M.J.: Polymake: a framework for analyzing convex polytopes. Kalai, G., Ziegler, G.M., editors. Polytopes–Combinatorics and Computation, pp. 43–74. Birkhäuser,Basel (2000)Google Scholar
  11. Herrmann, S., Joswig, M., Speyer, D.: Dressians, tropical Grassmannians, and their rays. Forum Mathematicum 26(6), 1853–1881 (2012)Google Scholar
  12. Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Combin. 16(2), pp. 26 (2009)Google Scholar
  13. Postnikov, A.: Total positivity, Grassmannians, and networks, pp. 76 (2006) (preprint). arXiv:math/0609764
  14. Speyer, D., Williams, L.: The tropical totally positive Grassmannian. J. Algebraic Combin. 22(2), 189–210 (2005)Google Scholar
  15. Speyer, D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008). doi: 10.1137/080716219
  16. Speyer, D.E.: A matroid invariant via the \(K\)-theory of the Grassmannian. Adv. Math. 221(3), 882–913 (2009). doi: 10.1016/j.aim.2009.01.010
  17. Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)Google Scholar
  18. Stanley, R.P., Pitman, J.: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Comput. Geom. 27(4), 603–634 (2002)Google Scholar
  19. Stein, W.A. et al.: Sage Mathematics Software (Version 6.8). The Sage Development Team, USA (2015).

Copyright information

© The Managing Editors 2016

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität BerlinBerlinGermany
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria
  3. 3.Einstein Institute of MathematicsHebrew University of JerusalemJerusalemIsrael

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