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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2020 Volume 16, 115, 8 pp. (Mi sigma1653)

The Full Symmetric Toda Flow and Intersections of Bruhat Cells

Yuri B. Chernyakovabc, Georgy I. Sharyginbda, Alexander S. Sorinbef, Dmitry V. Talalaevdga

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

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.

Keywords: Lie groups, Bruhat order, integrable systems, Toda flow.

MSC: 22E15, 70H06

Received: July 13, 2020; in final form November 2, 2020; Published online November 11, 2020

Language: English

DOI: 10.3842/SIGMA.2020.115



Bibliographic databases:
ArXiv: 1810.09622


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