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

Symmetry Integrability and Geometry-Methods and Applications

article

Yury B. Chernyakov¹  Georgy I. Sharygin²  Alexander S. Sorin³ 

[1] Institute for Theoretical and Experimental Physics;Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics;Lomonosov Moscow State University, Faculty of Mechanics and Mathema

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

DOI  :  10.3842/SIGMA.2016.084

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown