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Cluster algebras of type $$D_4$$, tropical planes, and the positive tropical Grassmannian. (English) Zbl 1401.13063
Summary: 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.

##### MSC:
 13F60 Cluster algebras 14T05 Tropical geometry (MSC2010) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
##### Software:
polymake; SageMath
Full Text:
##### References:
 [1] Ceballos, C., Labbé, J.-P., Stump, C.: Subword complexes, cluster complexes, and generalized multi-associahedra. J. Algebr. Combin. 39(1), 17-51 (2014) · Zbl 1286.05180 [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) · Zbl 1350.13020 [3] Ceballos, C., Pilaud, V.: Cluster algebras of type D: pseudotriangulations approach. Electron. J. Combin. 22(4), pp. 27 (2015b) · Zbl 1350.13021 [4] Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83-146 (2008) · Zbl 1263.13023 [5] Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002) · Zbl 1021.16017 [6] Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. (2), 158(3), 977-1018 (2003a) · Zbl 1057.52003 [7] Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63-121 (2003b) · Zbl 1054.17024 [8] Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1-52 (2005) · Zbl 1135.16013 [9] Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112-164 (2007) · Zbl 1127.16023 [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) · Zbl 0960.68182 [11] Herrmann, S., Joswig, M., Speyer, D.: Dressians, tropical Grassmannians, and their rays. Forum Mathematicum 26(6), 1853-1881 (2012) · Zbl 1308.14068 [12] Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Combin. 16(2), pp. 26 (2009) · Zbl 1195.14080 [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) · Zbl 1094.14048 [15] Speyer, D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527-1558 (2008). doi:10.1137/080716219 · Zbl 1191.14076 [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 · Zbl 1222.14131 [17] Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389-411 (2004) · Zbl 1065.14071 [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) · Zbl 1012.52019 [19] Stein, W.A. et al.: Sage Mathematics Software (Version 6.8). The Sage Development Team, USA (2015). http://www.sagemath.org
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