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

SIGMA 16 (2020), 115, 8 pages      arXiv:1810.09622      https://doi.org/10.3842/SIGMA.2020.115

The Full Symmetric Toda Flow and Intersections of Bruhat Cells

Yuri B. Chernyakov ^abc, Georgy I. Sharygin ^abd, Alexander S. Sorin ^bef and Dmitry V. Talalaev ^adg
a) Institute for Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia
^b) Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Moscow region, Russia
^c) Institute for Information Transmission Problems, Bolshoy Karetny per.19, build. 1, 127994, Moscow, Russia
^d) Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, GSP-1, 1 Leninskiye Gory, Main Building, 119991 Moscow, Russia
^e) National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, 115409 Moscow, Russia
^f) Dubna State University, 141980 Dubna, Moscow region, Russia
^g) Centre of integrable systems, P.G. Demidov Yaroslavl State University, 150003, 14 Sovetskaya Str., Yaroslavl, Russia

Received July 13, 2020, in final form November 02, 2020; Published online November 11, 2020

Abstract
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.

Key words: Lie groups; Bruhat order; integrable systems; Toda flow.

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