Geodesic
Phase portraits of the full symmetric Toda systems on rank-$2$ groups
Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 2, pp. 193-213 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank $2$. For this, we verify this property for the two remaining rank-$2$ groups, $Sp(4,\mathbb R)$ and the real form of $G_2$.
Keywords: full symmetric Toda system, Bruhat order, Morse function, Weyl group.
Mots-clés : semisimple Lie group 
@article{TMF_2017_193_2_a1,
     author = {A. S. Sorin and Yu. B. Chernyakov and G. I. Sharygin},
     title = {Phase portraits of the~full symmetric {Toda} systems on rank-$2$ groups},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {193--213},
     year = {2017},
     volume = {193},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a1/}
}
TY  - JOUR
AU  - A. S. Sorin
AU  - Yu. B. Chernyakov
AU  - G. I. Sharygin
TI  - Phase portraits of the full symmetric Toda systems on rank-$2$ groups
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 193
EP  - 213
VL  - 193
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a1/
LA  - ru
ID  - TMF_2017_193_2_a1
ER  - 
%0 Journal Article
%A A. S. Sorin
%A Yu. B. Chernyakov
%A G. I. Sharygin
%T Phase portraits of the full symmetric Toda systems on rank-$2$ groups
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 193-213
%V 193
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a1/
%G ru
%F TMF_2017_193_2_a1
A. S. Sorin; Yu. B. Chernyakov; G. I. Sharygin. Phase portraits of the full symmetric Toda systems on rank-$2$ groups. Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 2, pp. 193-213. http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a1/

[1] A. M. Bloch, R. W. Brockett, T. S. Ratiu, “Completely integrable gradient flows”, Commun. Math. Phys., 147:1 (1992), 57–74 | DOI | MR | Zbl

[2] Y. Kodama, J. Ye, “Iso-spectral deformations of general matrix and their reductions on Lie algebras”, Commun. Math. Phys., 178:3 (1996), 765–788 | DOI | MR | Zbl

[3] Yu. B. Chernyakov, G. I. Sharygin, A. S. Sorin, “Bruhat order in full symmetric Toda system”, Commun. Math. Phys., 330:1 (2014), 367–399, arXiv: 1212.4803 | DOI | MR | Zbl

[4] Yu. B. Chernyakov, G. I. Sharygin, A. S. Sorin, “Bruhat order in the full symmetric $\mathfrak{sl}_n$ Toda lattice on partial flag space”, SIGMA, 12 (2016), 084, 25 pp., arXiv: 1412.8116 | MR | Zbl

[5] P. Deift, T. Nanda, C. Tomei, “Ordinary differential equations and the symmetric eigenvalue problem”, SIAM J. Numer. Anal., 20:1 (1983), 1–22 | DOI | MR | Zbl

[6] P. Fré, A. S. Sorin, “The Weyl group and asymptotics: All supergravity billiards have a closed form general integral”, Nucl. Phys. B, 815:3 (2009), 430–494, arXiv: 0710.1059 | DOI | MR | Zbl

[7] P. Fré, A. S. Sorin, M. Trigiante, “Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits”, JHEP, 04 (2012), 015, 105 pp., arXiv: 1103.0848 | MR

[8] Y. Kodama, L. Williams, “The full Kostant–Toda hierarchy on the positive flag variety”, Commun. Math. Phys., 335:1 (2015), 247–283, arXiv: 1308.5011 | DOI | MR

[9] U. Fulton, Tablitsy Yunga i ikh prilozheniya k teorii predstavlenii i geometrii, MTsNMO, M., 2006 | MR | Zbl

[10] A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005 | MR

[11] M. Brion, “Lectures on the Geometry of Flag Varieties”, Topics in Cohomological Studies of Algebraic Varieties, Trends in Mathematics, ed. P. Pragacz, Birkhäuser, Basel, 2005, 33–85, arXiv: math/0410240 | DOI | MR

[12] F. De Mari, M. Pedroni, “Toda flows and real Hessenberg manifolds”, J. Geom. Anal., 9:4 (1999), 607–625 | DOI | MR | Zbl

[13] P. Deift, L. C. Li, T. Nanda, C. Tomei, “The Toda flow on a generic orbit is integrable”, Commun. Pure Appl. Math., 39:2 (1986), 183–232 | DOI | MR | Zbl

[14] N. Ercolani, H. Flaschka, S. Singer, “The geometry of the full Kostant–Toda lattice”, Integrable Systems (Luminy, France, July 1–5, 1991), Progress in Mathematics, 115, eds. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, Birkhäuser, Boston, 1993, 181–226 | MR

[15] M. Adler, “On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg–Devries type equations”, Invent. Math., 50:3 (1979), 219–248 | DOI | MR | Zbl

[16] B. Kostant, “The solution to a generalized Toda lattice and representation theory”, Adv. Math., 34:3 (1979), 195–338 | DOI | MR | Zbl

[17] W. W. Symes, “Systems of Toda type, inverse spectral problems, and representation theory”, Invent. Math., 59:1 (1980), 13–51 | DOI | MR | Zbl

[18] Yu. B. Chernyakov, A. S. Sorin, “Explicit semi-invariants and integrals of the full symmetric $\mathfrak{sl}_n$ Toda lattice”, Lett. Math. Phys., 104:8 (2014), 1045–1052, arXiv: 1306.1647 | DOI | MR | Zbl

[19] K. I. Gross, “The Plancherel transform on the nilpotent part of $G_2$ and some applications to the representation theory of $G_2$”, Trans. Amer. Math. Soc., 132:2 (196), 411–446 | DOI | MR

[20] A. S. Sorin, Yu. B. Chernyakov, “Novyi metod postroeniya poluinvariantov i integralov polnoi simmetrichnoi $\mathfrak{sl}_n$ reshetki Tody”, TMF, 183:2 (2015), 222–253, arXiv: 1312.4555 | DOI | DOI | MR