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SIGMA 11 (2015), 033, 32 pages arXiv:1409.8622
https://doi.org/10.3842/SIGMA.2015.033
Contribution to the Special Issue on New Directions in Lie Theory
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
Yuki Kanakubo and Toshiki Nakashima
Division of Mathematics, Sophia University, Yonban-cho 4, Chiyoda-ku, Tokyo 102-0081, Japan
Received October 01, 2014, in final form April 14, 2015; Published online April 23, 2015
Abstract
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 ˉA(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.
Key words:
cluster variables; double Bruhat cells; crystal bases; monomial realizations, generalized minors.
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