Geodesic
Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $G^{u,v}=BuB\cap B_-vB_-$ 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]. In the case $G={\rm SL}_{r+1}({\mathbb C})$, $v=e$ 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.
Keywords: cluster variables; double Bruhat cells; crystal bases; monomial realizations, generalized minors. 
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     author = {Yuki Kanakubo and Toshiki Nakashima},
     title = {Cluster {Variables} on {Certain} {Double} {Bruhat} {Cells} of {Type} $(u,e)$ and {Monomial} {Realizations} of {Crystal} {Bases} of {Type~A}},
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Yuki Kanakubo; Toshiki Nakashima. Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a32/

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