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Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
Інший ID:
2010 Mathematics Subject Classification: 13F60; 81R50; 17B37
DOI:10.3842/SIGMA.2015.033
DOI:10.3842/SIGMA.2015.033
Посилання:
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.
Підтримка:
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.
Дата:
2015
Переглядів:
619
Завантажень:
222
Анотація:
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.
