Monomial realizations and LS paths of fundamental representations for rank 2 Kac-Moody algebras

For a Kac-Moody algebra $g$ of rank 2 and a fundamental weight λ, we explicitly give an isomorphism between the set of Lakshmibai-Seshadri paths $B\left(\lambda \right)$ and monomial realization $M\left(\lambda \right)$. As an application, we also give an explicit form of monomial realization $M\left(\lambda \right)$ in terms of Weyl groups.