Yu. B. Chernyakov, G. I. Sharygin, A. S. Sorin
In this paper we discuss some geometric and topologiacal properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagramm for the full symmetric Toda system in dimensions n=3,4 coincides with the Hasse diagramm of the Bruhat order of symmetric group 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 extand 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 diagramm. This is equivalent to the fact, that the system at hand is in fact a Morse-Smale system.
View original:
http://arxiv.org/abs/1212.4803
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