The Full Symmetric Toda Flow and Intersections of Bruhat Cells
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 , ′ in the Weyl group (), the corresponding real Bruhat cell intersects with the dual Bruhat cell ′ iff ≺ ′ in the Bruhat order o...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211005 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862532624048717824 |
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| author | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| author_facet | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| citation_txt | The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 , ′ in the Weyl group (), the corresponding real Bruhat cell intersects with the dual Bruhat cell ′ iff ≺ ′ in the Bruhat order on (). Here is a normal real form of a semisimple complex Lie algebra 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.
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| first_indexed | 2026-03-12T14:01:24Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211005 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:01:24Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. 2025-12-22T09:24:28Z 2020 The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages 1815-0659 2020 Mathematics Subject Classification: 22E15; 70H06 arXiv:1810.09622 https://nasplib.isofts.kiev.ua/handle/123456789/211005 https://doi.org/10.3842/SIGMA.2020.115 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 , ′ in the Weyl group (), the corresponding real Bruhat cell intersects with the dual Bruhat cell ′ iff ≺ ′ in the Bruhat order on (). Here is a normal real form of a semisimple complex Lie algebra 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. The authors thank Sergey Loktev, Vladimir Rubtsov, and Evgeniy Smirnov for useful discussions. The authors would like to express their gratitude to the anonymous referee, whose remarks have helped us to improve the presentation of this paper. The work of Yu.B. Chernyakov was partly supported by the grant RFBR-18-02-01081. The work of G.I. Sharygin was partly supported by the grant RFBR-18-01-00398. The work of D.V. Talalaev was partly supported by the grant Leader(math) 20-7-1-21-1 of the foundation for the advancement of theoretical physics and mathematics BASIS and within the framework of a development program for the Regional Scienti c and Educational Mathematical Center of the Yaroslavl State University with nancial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2020-1514/1 additional to the agreement on provision of subsidies from the federal budget No. 075-02-2020-1514). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Full Symmetric Toda Flow and Intersections of Bruhat Cells Article published earlier |
| spellingShingle | The Full Symmetric Toda Flow and Intersections of Bruhat Cells Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| title | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_full | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_fullStr | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_full_unstemmed | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_short | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_sort | full symmetric toda flow and intersections of bruhat cells |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211005 |
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