Geodesic
The Full Symmetric Toda Flow and Intersections of Bruhat Cells
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

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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. 
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/
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[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