Geodesic
The Full Symmetric Toda Flow and Intersections of Bruhat Cells
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.
Keywords: Lie groups, Bruhat order, integrable systems, Toda flow. 
@article{SIGMA_2020_16_a114,
     author = {Yuri B. Chernyakov and Georgy I. Sharygin and Alexander S. Sorin and Dmitry V. Talalaev},
     title = {The {Full} {Symmetric} {Toda} {Flow} and {Intersections} of {Bruhat} {Cells}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a114/}
}
TY  - JOUR
AU  - Yuri B. Chernyakov
AU  - Georgy I. Sharygin
AU  - Alexander S. Sorin
AU  - Dmitry V. Talalaev
TI  - The Full Symmetric Toda Flow and Intersections of Bruhat Cells
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2020
VL  - 16
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a114/
LA  - en
ID  - SIGMA_2020_16_a114
ER  - 
%0 Journal Article
%A Yuri B. Chernyakov
%A Georgy I. Sharygin
%A Alexander S. Sorin
%A Dmitry V. Talalaev
%T The Full Symmetric Toda Flow and Intersections of Bruhat Cells
%J Symmetry, integrability and geometry: methods and applications
%D 2020
%V 16
%U http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a114/
%G en
%F SIGMA_2020_16_a114
Yuri B. Chernyakov; Georgy I. Sharygin; Alexander S. Sorin; Dmitry V. Talalaev. The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a114/

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

[2] Bloch A. M., Gekhtman M. I., “Hamiltonian and gradient structures in the Toda flows”, J. Geom. Phys., 27 (1998), 230–248 | DOI | MR | Zbl

[3] Brion M., Lakshmibai V., “A geometric approach to standard monomial theory”, Represent. Theory, 7 (2003), 651–680, arXiv: math.AG/0111054 | DOI | MR | Zbl

[4] Casian L., Kodama Y., “Toda lattice, cohomology of compact Lie groups and finite Chevalley groups”, Invent. Math., 165 (2006), 163–208, arXiv: math.AT/0504329 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] Chernyakov Yu.B., Sharygin G. I., Sorin A. S., “Bruhat order in the Toda system on $\mathfrak{so}(2,4)$: an example of non-split real form”, J. Geom. Phys., 136 (2019), 45–51, arXiv: 1712.0913 | DOI | MR | Zbl

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

[8] Deodhar V. V., “On some geometric aspects of Bruhat orderings. I A finer decomposition of Bruhat cells”, Invent. Math., 79 (1985), 499–511 | DOI | MR | Zbl

[9] Faybusovich L., “Toda flows and isospectral manifolds”, Proc. Amer. Math. Soc., 115 (1992), 837–847 | DOI | MR | Zbl

[10] Fulton W., Young tableaux with applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[11] Sorin A. S., Chernyakov Yu.B., Sharygin G. I., “Phase portrait of the full symmetric Toda system on rank-2 groups”, Theoret. and Math. Phys., 193 (2017), 1574–1592, arXiv: 1512.05821 | DOI | MR | Zbl