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 SL m (ℂ).
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Alexeev, V., Brion, M.: Toric degenerations of spherical varieties. Sel. Math. 10(4), 453–478 (2005)
Bocci, C., Cools, F.: A tropical interpretation of m-dissimilarity maps. Appl. Math. Comput. 212(2), 349–356 (2009)
Cools, F.: On the relation between weighted trees and tropical Grassmannians. J. Symb. Comput. 44(8), 1079–1086 (2009)
Dolgachev, I.: Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003)
Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991)
Giraldo, B.I.: Dissimilarity vectors of trees are contained in the tropical Grassmannian. Electron. J. Comb. 17(1) (2010)
Grosshans, F.D.: Algebraic Homogeneous Spaces and Invariant Theory. Springer Lecture Notes, vol. 1673. Springer, Berlin (1997)
Manon, C.: The algebra of conformal blocks. http://arxiv.org/abs/0910.0577
Manon, C.: Graded valuations and tropical geometry. http://arxiv.org/abs/1006.0038
Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16(3), 543–556 (2009)
Pachter, L., Sturmfels, B.: Algebraic Statistics for Computational Biology. Cambridge University Press, New York (2005)
Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)
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This work was supported by the NSF fellowship DMS-0902710.
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Manon, C. Dissimilarity maps on trees and the representation theory of SL m (ℂ). J Algebr Comb 33, 199–213 (2011). https://doi.org/10.1007/s10801-010-0241-9
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DOI: https://doi.org/10.1007/s10801-010-0241-9