Communications in Mathematical Physics

, Volume 330, Issue 1, pp 367–399 | Cite as

Bruhat Order in Full Symmetric Toda System

  • Yu. B. ChernyakovEmail author
  • G. I. Sharygin
  • A. S. Sorin


In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions n = 3, 4 coincides with the Hasse diagram of the Bruhat order of symmetric groups S 3 and S 4. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general n. The resulting theorem identifies the set of singular points of dim = n Toda flow with the elements of the permutation group S n , so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagram. This is equivalent to the fact that the full symmetric Toda system is in fact a Morse–Smale system.


Singular Point Phase Portrait Weyl Group Morse Theory Morse Function 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Yu. B. Chernyakov
    • 1
    • 2
    Email author
  • G. I. Sharygin
    • 1
    • 2
    • 3
    • 4
  • A. S. Sorin
    • 2
  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubnaRussia
  3. 3.Faculty of Mechanics and Mathematics, Lomonosov Moscow State UniversityMoscowRussia
  4. 4.Laboratory of Discrete and Computational GeometryYaroslavl’ State UniversityYaroslavl’Russia

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