Geodesic
Bruhat Order in the Full Symmetric $\mathfrak{sl}_n$ Toda Lattice on Partial Flag Space
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: full symmetric Toda lattice; Bruhat order; integrals and semi-invariants; partial flag space; Morse function; multiset permutation. 
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     author = {Yury B. Chernyakov and Georgy I. Sharygin and Alexander S. Sorin},
     title = {Bruhat {Order} in the {Full} {Symmetric} $\mathfrak{sl}_n$ {Toda} {Lattice} on {Partial} {Flag} {Space}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a83/}
}
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Yury B. Chernyakov; Georgy I. Sharygin; Alexander S. Sorin. Bruhat Order in the Full Symmetric $\mathfrak{sl}_n$ Toda Lattice on Partial Flag Space. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a83/

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