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

SIGMA 12 (2016), 084, 25 pages      arXiv:1412.8116

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

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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  1. Adler M., On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math. 50 (1978), 219-248.
  2. Arhangel'skiǐ A.A., Completely integrable Hamiltonian systems on the group of triangular matrices, Math. USSR Sb. 36 (1980), 127-134.
  3. Björner A., Brenti F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005.
  4. Brion M., Lectures on the geometry of flag varieties, math.AG/0410240.
  5. Chernyakov Yu.B., Sharygin G.I., Sorin A.S., Bruhat order in full symmetric Toda system, Comm. Math. Phys. 330 (2014), 367-399, arXiv:1212.4803.
  6. Chernyakov Yu.B., Sorin A.S., Explicit semi-invariants and integrals of the full symmetric $\mathfrak{sl}_n$ Toda lattice, Lett. Math. Phys. 104 (2014), 1045-1052, arXiv:1306.1647.
  7. De Mari F., Pedroni M., Toda flows and real Hessenberg manifolds, J. Geom. Anal. 9 (1999), 607-625.
  8. Deift P., Li L.C., Nanda T., Tomei C., The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39 (1986), 183-232.
  9. Deift P., Nanda T., Tomei C., Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal. 20 (1983), 1-22.
  10. Ercolani N.M., Flaschka H., Singer S., The geometry of the full Kostant-Toda lattice, in Integrable Systems (Luminy, 1991), Progr. Math., Vol. 115, Birkhäuser Boston, Boston, MA, 1993, 181-225.
  11. Flaschka H., The Toda lattice. I. Existence of integrals, Phys. Rev. B 9 (1974), 1924-1925.
  12. Flaschka H., On the Toda lattice. II. Inverse-scattering solution, Progr. Theoret. Phys. 51 (1974), 703-716.
  13. Fré P., Sorin A.S., The Weyl group and asymptotics: all supergravity billiards have a closed form general integral, Nuclear Phys. B 815 (2009), 430-494, arXiv:0710.1059.
  14. Fulton W., Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
  15. Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.
  16. Hénon M., Integrals of the Toda lattice, Phys. Rev. B 9 (1974), 1921-1923.
  17. Kodama Y., McLaughlin K.T.-R., Explicit integration of the full symmetric Toda hierarchy and the sorting property, Lett. Math. Phys. 37 (1996), 37-47, solv-int/9502006.
  18. Kodama Y., Williams L., The full Kostant-Toda hierarchy on the positive flag variety, Comm. Math. Phys. 335 (2015), 247-283, arXiv:1308.5011.
  19. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. Math. 34 (1979), 195-338.
  20. Moser J., Finitely many mass points on the line under the influence of an exponential potential - an integrable system, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, 467-497.
  21. Symes W.W., Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math. 59 (1980), 13-51.
  22. Toda M., Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431-436.
  23. Toda M., Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967), 501-506.

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