Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized min...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147010 |
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| Cite this: | Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ. |
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Kanakubo, Y. Nakashima, T. 2019-02-12T20:32:15Z 2019-02-12T20:32:15Z 2015 Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 81R50; 17B37 DOI:10.3842/SIGMA.2015.033 https://nasplib.isofts.kiev.ua/handle/123456789/147010 Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized minors {Δ(k;i)} are the cluster variables belonging to a given initial seed in C[Gu,v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case G=SLr₊₁(C), v=e and some special u∈W, we shall describe the generalized minors {Δ(k;i)} as summations of monomial realizations of certain Demazure crystals. This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html. The authors would like to acknowledge the referees for giving them relevant advice and suggestion to improve this article. T.N. is supported in part by JSPS Grants in Aid for Scientific Research ]22540031, ]15K04794. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A |
| spellingShingle |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A Kanakubo, Y. Nakashima, T. |
| title_short |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A |
| title_full |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A |
| title_fullStr |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A |
| title_full_unstemmed |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A |
| title_sort |
cluster variables on certain double bruhat cells of type (u,e) and monomial realizations of crystal bases of type a |
| author |
Kanakubo, Y. Nakashima, T. |
| author_facet |
Kanakubo, Y. Nakashima, T. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized minors {Δ(k;i)} are the cluster variables belonging to a given initial seed in C[Gu,v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case G=SLr₊₁(C), v=e and some special u∈W, we shall describe the generalized minors {Δ(k;i)} as summations of monomial realizations of certain Demazure crystals.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147010 |
| citation_txt |
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ. |
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2025-12-07T16:21:51Z |
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2025-12-07T16:21:51Z |
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