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The tropical totally positive Grassmannians. (English) Zbl 1094.14048
The additive group \((\mathbb{R},+)\) together with the binary operation “min” is a semiring, called the tropical semiring. The geometric objects of tropical geometry are tropical varieties, polyhedral cell complexes that are contained in tropical affine spaces or in tropical projective spaces over the tropical semiring. The field \(K\) of Puiseux series with coefficients in \(\mathbb{C}\) carries the order valuation, which takes its values in the tropical semiring. The order valuation is used to associate a tropical variety, the tropicalization, with every algebraic variety \(V(I)\subseteq(K\setminus\{0 \})^n\). The authors introduce the positive part of the tropicalization of an affine variety and study the notion for the case of Grassmannians. Tropicalizations and their positive parts are closed subcomplexes of the Gröbner complex associated with the defining ideal \(I\). The positive parts of several tropicalized Grassmannians are closely related to a class of polytopes, called associahedra. The connection is described explicitly.

14P99 Real algebraic and real-analytic geometry
16Y60 Semirings
14M15 Grassmannians, Schubert varieties, flag manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
12J10 Valued fields
52B70 Polyhedral manifolds
polymake; cdd
Full Text: DOI arXiv
[1] Chapoton, F.; Fomin, S.; Zelevinsky, A., Polytopal realizations of generalized associahedra, Canadian Mathematical Bulletin, 45, 537-566, (2002) · Zbl 1018.52007
[2] Fomin, S.; Zelevinsky, A., Cluster algebras I: Foundations, Journal of the AMS, 15, 497-529, (2002) · Zbl 1021.16017
[3] Fomin, S.; Zelevinsky, A., Cluster algebras II: Finite type classification, Inventiones Mathematicae, 154, 63-121, (2003) · Zbl 1054.17024
[4] Fomin, S.; Zelevinsky, A., Total positivity: Tests and parameterizations, The Mathematical Intelligencer, 22, 23-33, (2000) · Zbl 1052.15500
[5] Fomin, S.; Zelevinsky, A., Y-systems and generalized associahedra, Annals of Mathematics, 158, 977-1018, (2003) · Zbl 1057.52003
[6] K. Fukuda, CDD-A C-implementation of the double description method, available from http://www.cs.mcgill.ca/∼fukuda/soft/cdd_home/cdd.html.
[7] E. Gawrilow and M. Joswig, polymake, available from http://www.math.tu-berlin.de/polymake.
[8] Gessel, I.; Viennot, G., Binomial determinants, paths and hook length formulae, Adv. In Math., 58, 300-321, (1985) · Zbl 0579.05004
[9] D. Handelman, “Positive polynomials and product type actions of compact groups,” Mem. Amer. Math. Soc. 320 (1985). · Zbl 0571.46045
[10] G. Lusztig, “Introduction to total positivity,” in “Positivity in Lie theory: Open problems,” (Ed.) J. Hilgert, J.D. Lawson, K.H. Neeb and E.B. Vinberg (Eds.), de Gruyter Berlin, 1998, pp. 133-145. · Zbl 0929.20035
[11] Lusztig, G., Total positivity in partial flag manifolds, Representation Theory, 2, 70-78, (1998) · Zbl 0895.14014
[12] G. Lusztig, “Total positivity in reductive groups,” in Lie theory and geometry: In honor of Bertram Kostant, Progress in Mathematics 123, Birkhauser, 1994, 531-568. · Zbl 0845.20034
[13] Einsiedler, M.; Tuncel, S., “When does a polynomial ideal contain a positive polynomial?,” in Effective Methods in Algebraic Geometry, J. Pure Appl. Algebra, 164, 149-152, (2001) · Zbl 0999.13009
[14] Mikhalkin, G., Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris, 336, 629-634, (2003) · Zbl 1027.14026
[15] A. Postnikov, “Webs in totally positive Grassman cells,” in preparation.
[16] J. Richter-Gebert, B. Sturmfels, and T. Theobald, “First steps in tropical geometry,” 29 pages, http://www.arxiv.org/math.AG/030636, to appear in “Idempotent Mathematics and Mathematical Physics”, (eds. G. Litvinov and V. Maslov), Proceedings Vienna, 2003. · Zbl 1093.14080
[17] J. Scott, “Grassmannians and cluster algebras,” preprint, http://www.arxiv.org/math.CO/031114 · Zbl 1088.22009
[18] Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 389-411, (2004) · Zbl 1065.14071
[19] Stanley, R.; Pitman, J., A polytope related to empirical distribution, plane trees, parking functions, and the associahedron, Discrete Comput. Geometry, 27, 603-634, (2002) · Zbl 1012.52019
[20] B. Sturmfels, “Gröbner bases and convex polytopes,” American Mathematical Society, Providence, 1991.
[21] B. Sturmfels, “Solving systems of polynomial equations,” American Mathematical Society, Providence, 2002. · Zbl 1101.13040
[22] Williams, L., Enumeration of totally positive Grassmann cells, Adv. Math., 190, 319-342, (2005) · Zbl 1064.05150
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