Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric sln Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation grou...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2016 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2016
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147851 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space / Y.B. Chernyakov, G.I. Sharygin, A.S. Sorin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 23 назв. — англ. |
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Chernyakov, Y.B. Sharygin, G.I. Sorin, A.S. 2019-02-16T09:17:24Z 2019-02-16T09:17:24Z 2016 Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space / Y.B. Chernyakov, G.I. Sharygin, A.S. Sorin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 06A06; 37D15; 37J35 DOI:10.3842/SIGMA.2016.084 https://nasplib.isofts.kiev.ua/handle/123456789/147851 In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric sln Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when t→±∞ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations. The authors would like to thank G. Koshevoy for the fruitful discussion. We also would like to thank the referees, whose remarks helped in a great measure to improve the paper. The work of Yu.B. Chernyakov was supported by grant RFBR-15-01-08462. The work of G.I. Sharygin was supported by grant RFBR-15-01-05990. The work of A.S. Sorin was partially supported by RFBR grants 15-52-05022-Arm-a and 16-52-12012-NNIO-a. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space |
| spellingShingle |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space Chernyakov, Y.B. Sharygin, G.I. Sorin, A.S. |
| title_short |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space |
| title_full |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space |
| title_fullStr |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space |
| title_full_unstemmed |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space |
| title_sort |
bruhat order in the full symmetric sln toda lattice on partial flag space |
| author |
Chernyakov, Y.B. Sharygin, G.I. Sorin, A.S. |
| author_facet |
Chernyakov, Y.B. Sharygin, G.I. Sorin, A.S. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric sln Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when t→±∞ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147851 |
| citation_txt |
Bruhat Order in the Full Symmetric sln Toda Lattice on Partial Flag Space / Y.B. Chernyakov, G.I. Sharygin, A.S. Sorin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 23 назв. — англ. |
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2025-12-07T17:59:37Z |
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2025-12-07T17:59:37Z |
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