Cluster algebras and Grassmannians of type G2

We prove a conjecture of Geiss, Leclerc and Schr\"{o}er, producing cluster algebra structures on multi-homogeneous coordinate ring of partial flag varieties, for the case $G_2$. As a consequence we sharpen the known fact that coordinate ring of the double Bruhat cell $G^{e,w_0}$ is an upper cluster algebra, by proving that it is a cluster algebra.