Cluster Variables on Double Bruhat Cells G^u,e of Classical Groups and Monomial Realizations of Demazure Crystals

2016/04/20 by Kanakubo, Yuki, Nakashima, Toshiki
#FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1604.05956

Abstract

Let G be a simply connected simple algebraic group over ℂ, B and B_- its two opposite Borel subgroups, and W the associated Weyl group. It is shown that the coordinate ring \mathbb C[G^u,v] (u, v∈ W) of the double Bruhat cell G^u,v=BuB∩ B_-vB_- is isomorphic to the cluster algebra A(i)_{\mathbb C} and the initial cluster variables of \mathbb C[G^u,v] are the generalized minors Δ(k;i) by Berenstein, Fomin, Zelevinsky, Goodearl and Yakimov. In the case that a classical group G is of type \rm B_r, \rm C_r or \rm D_r, we shall describe the non-trivial last r initial cluster variables \Δ(k;i)\_(m-2)r

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