The Tropical Totally Positive Grassmannian
David Speyer1
and Lauren Williams2
1Department of Mathematics University of California Berkeley
2Department of Mathematics MIT Cambridge
2Department of Mathematics MIT Cambridge
DOI: 10.1007/s10801-005-2513-3
Abstract
Tropical algebraic geometry is the geometry of the tropical semiring (\Bbb R, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop + Gr k, n. We show that Trop + Gr 2, n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type A n - 3) associahedron, and that Trop + Gr 3,6 and Trop + Gr 3,7 are closely related to the fans dual to the types D 4 and E 6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types A n - 3, D 4, and E 6 for Gr 2, n, Gr 3,6, and Gr 3,7. We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.
Pages: 189–210
Keywords: keywords tropical geometry; total positivity; cluster algebras; generalized associahedra; Grassmannian
Full Text: PDF
References
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15. A. Postnikov, “Webs in totally positive Grassman cells,” in preparation.
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17. J. Scott, “Grassmannians and cluster algebras,” preprint, htt p : //www.ar xiv.org/math.C O/0311148
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19. R. Stanley and J. Pitman, “A polytope related to empirical distribution, plane trees, parking functions, and the associahedron,” Discrete Comput. Geometry 27 (2002), 603-634. SPEYER AND WILLIAMS
20. B. Sturmfels, “Gr\ddot obner bases and convex polytopes,” American Mathematical Society, Providence, 1991.
21. B. Sturmfels, “Solving systems of polynomial equations,” American Mathematical Society, Providence, 2002.
22. L. Williams, “Enumeration of totally positive Grassmann cells,” Adv. Math. 190 (2005), 319-342.
2. S. Fomin and A. Zelevinsky, “Cluster algebras I: Foundations,” Journal of the AMS 15 (2002), 497- 529.
3. S. Fomin and A. Zelevinsky, “Cluster algebras II: Finite type classification,” Inventiones Mathematicae 154 (2003), 63-121.
4. S. Fomin and A. Zelevinsky, “Total positivity: Tests and parameterizations,” The Mathematical Intelligencer 22 (2000), 23-33.
5. S. Fomin and A. Zelevinsky, “Y-systems and generalized associahedra,” Annals of Mathematics 158(3) (2003), 977-1018.
6. K. Fukuda, CDD-A C-implementation of the double description method, available from http://www. cs.mcgill.ca/\sim fukuda/soft/cdd home/cdd.html.
7. E. Gawrilow and M. Joswig, polymake, available from http://www.math.tu-berlin.de/polymake/.
8. I. Gessel and G. Viennot, “Binomial determinants, paths and hook length formulae,” Adv. In Math. 58 (1985), 300-321.
9. D. Handelman, “Positive polynomials and product type actions of compact groups,” Mem. Amer. Math. Soc. 320 (1985).
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.
11. G. Lusztig, “Total positivity in partial flag manifolds,” Representation Theory, 2 (1998), 70-78.
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.
13. M. Einsiedler and S. Tuncel, “When does a polynomial ideal contain a positive polynomial?,” in Effective Methods in Algebraic Geometry, J. Pure Appl. Algebra 164(1/2) (2001), 149-152.
14. G. Mikhalkin, “Counting curves via lattice paths in polygons,” C. R. Math. Acad. Sci. Paris 336 (2003), 629-634.
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/0306366, to appear in “Idempotent Mathematics and Mathematical Physics”, (eds. G. Litvinov and V. Maslov), Proceedings Vienna, 2003.
17. J. Scott, “Grassmannians and cluster algebras,” preprint, htt p : //www.ar xiv.org/math.C O/0311148
18. D. Speyer and B. Sturmfels, “The tropical Grassmannian,” Adv. Geom. 4 (2004), 389-411.
19. R. Stanley and J. Pitman, “A polytope related to empirical distribution, plane trees, parking functions, and the associahedron,” Discrete Comput. Geometry 27 (2002), 603-634. SPEYER AND WILLIAMS
20. B. Sturmfels, “Gr\ddot obner bases and convex polytopes,” American Mathematical Society, Providence, 1991.
21. B. Sturmfels, “Solving systems of polynomial equations,” American Mathematical Society, Providence, 2002.
22. L. Williams, “Enumeration of totally positive Grassmann cells,” Adv. Math. 190 (2005), 319-342.