Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 199–213

Dissimilarity maps on trees and the representation theory of SLm(ℂ)

Open Access
Article

Abstract

We prove that m-dissimilarity vectors of weighted trees are points on the tropical Grassmannian, as conjecture by Cools in response to a question of Sturmfels and Pachter. We accomplish this by relating m-dissimilarity vectors to the representation theory of SLm(ℂ).

Keywords

Tropical geometry Representation theory 

References

  1. 1.
    Alexeev, V., Brion, M.: Toric degenerations of spherical varieties. Sel. Math. 10(4), 453–478 (2005) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bocci, C., Cools, F.: A tropical interpretation of m-dissimilarity maps. Appl. Math. Comput. 212(2), 349–356 (2009) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cools, F.: On the relation between weighted trees and tropical Grassmannians. J. Symb. Comput. 44(8), 1079–1086 (2009) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dolgachev, I.: Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003) MATHCrossRefGoogle Scholar
  5. 5.
    Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991) MATHGoogle Scholar
  6. 6.
    Giraldo, B.I.: Dissimilarity vectors of trees are contained in the tropical Grassmannian. Electron. J. Comb. 17(1) (2010) Google Scholar
  7. 7.
    Grosshans, F.D.: Algebraic Homogeneous Spaces and Invariant Theory. Springer Lecture Notes, vol. 1673. Springer, Berlin (1997) MATHGoogle Scholar
  8. 8.
    Manon, C.: The algebra of conformal blocks. http://arxiv.org/abs/0910.0577
  9. 9.
    Manon, C.: Graded valuations and tropical geometry. http://arxiv.org/abs/1006.0038
  10. 10.
    Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16(3), 543–556 (2009) MATHMathSciNetGoogle Scholar
  11. 11.
    Pachter, L., Sturmfels, B.: Algebraic Statistics for Computational Biology. Cambridge University Press, New York (2005) MATHCrossRefGoogle Scholar
  12. 12.
    Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations