Journal of Algebraic Combinatorics

, Volume 30, Issue 2, pp 173–191 | Cite as

Matching polytopes, toric geometry, and the totally non-negative Grassmannian

  • Alexander Postnikov
  • David Speyer
  • Lauren Williams
Article

Abstract

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ΔG can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell ΔG we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety XG. We use our technology to prove that the cell decomposition of (Grk,n)≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Grk,n)≥0 is 1.

Keywords

Total positivity Grassmannian CW complexes Birkhoff polytope Matching Matroid polytope Cluster algebra 

References

  1. 1.
    Billera, L., Sarangarajan, A.: The Combinatorics of Permutation Polytopes. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 24 (1996) Google Scholar
  2. 2.
    Cox, D.: What is a toric variety? In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. math., vol. 334, pp. 203–223. Amer. Math. Soc., Providence (2003) Google Scholar
  3. 3.
    Fomin, S., Shapiro, M.: Stratified spaces formed by totally positive varieties. Mich. Math. J. 48, 253–270 (2000) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 22, 497–527 (2002) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  6. 6.
    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes and simplicial complexes. http://www.math.tu-berlin.de/polymake (1997–2003). Version 2.0, with contributions by Thilo Schroder and Nikolaus Witte
  7. 7.
    Gelfand, I., Goresky, R., MacPherson, R., Serganova, V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Lovasz, L., Plummer, M.: Matching Theory. Elsevier Science, New York (1986) MATHGoogle Scholar
  9. 9.
    Lusztig, G.: Introduction to total positivity. In: Hilgert, J., Lawson, J.D., Neeb, K.H., Vinberg, E.B. (eds.) Positivity in Lie Theory: Open Problems, pp. 133–145. de Gruyter, Berlin (1998) Google Scholar
  10. 10.
    Lusztig, G.: Total positivity in partial flag manifolds. Represent. Theory 2, 70–78 (1998) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lusztig, G.: Total positivity in reductive groups. In: Lie Theory and Geometry: in Honor of Bertram Kostant. Progress in Mathematics, vol. 123. Birkhauser, Basel (1994) Google Scholar
  12. 12.
    Marsh, R., Rietsch, K.: Parametrizations of flag varieties. Represent. Theory, 8 (2004) Google Scholar
  13. 13.
    Postnikov, A.: Total positivity, Grassmannians, and networks. http://front.math.ucdavis.edu/math.CO/0609764
  14. 14.
    Rietsch, K.: Total positivity and real flag varieties. PhD dissertation, MIT (1998) Google Scholar
  15. 15.
    Rietsch, K.: Closure relations for totally non-negative cells in G/P. Math. Res. Lett. 13(5–6), 775–786 (2006) MATHMathSciNetGoogle Scholar
  16. 16.
    Rietsh, K., Williams, L.: The totaly nonnegative part of G/p is a CW complex. Transformation Groups, to appear Google Scholar
  17. 17.
    Scott, J.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. 92(2), 345–380 (2006) MATHCrossRefGoogle Scholar
  18. 18.
    Sottile, F.: Toric ideals, real toric varieties, and the moment map. In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. Math., vol. 334, pp. 225–240 (2003) Google Scholar
  19. 19.
    Speyer, D., Williams, L.: The tropical totally positive Grassmannian. J. Algebr. Comb. 22(2), 189–210 (2005) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Talaska, K.: A formula for Plücker coordinates of a perfectly oriented network. arXiv:0801.4822; Int. Math. Res. Not., to appear
  21. 21.
    Williams, L.: Enumeration of totally positive Grassmann cells. Adv. Math. 190(2), 319–342 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Williams, L.: Shelling totally non-negative flag varieties. J. Reine Angew. Math. 609, 1–22 (2007) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Alexander Postnikov
    • 1
  • David Speyer
    • 1
  • Lauren Williams
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations