### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 084, 25 pages      arXiv:1412.8116      https://doi.org/10.3842/SIGMA.2016.084

### Bruhat Order in the Full Symmetric $\mathfrak{sl}_n$ Toda Lattice on Partial Flag Space

Yury B. Chernyakov ab, Georgy I. Sharygin abc and Alexander S. Sorin bde
a) Institute for Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya, 117218, Moscow, Russia
b) Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980, Dubna, Moscow region, Russia
c) Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, GSP-1, 1 Leninskiye Gory, Main Building, 119991, Moscow, Russia
d) National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoye Shosse, 115409 Moscow, Russia
e) Dubna International University, 141980, Dubna, Moscow region, Russia

Received February 15, 2016, in final form August 10, 2016; Published online August 20, 2016

Abstract
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations.

Key words: full symmetric Toda lattice; Bruhat order; integrals and semi-invariants; partial flag space; Morse function; multiset permutation.

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