Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A

Abstract

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_{-}$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$, $v\in W$, it is known that the coordinate ring $\mathbb{C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_{-}v{B}_{-}$ is isomorphic to an upper cluster algebra $\bar{\mathcal{A}}(\mathbf{i}{)}_{\mathbb{C}}$ and the generalized minors $\{\Delta (k;\mathbf{i})\}$ are the cluster variables belonging to a given initial seed in $\mathbb{C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $G={\mathrm{SL}}_{r+1}(\mathbb{C})$, $v=e$ and some special $u\in W$, we shall describe the generalized minors $\{\Delta (k;\mathbf{i})\}$ as summations of monomial realizations of certain Demazure crystals.