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(λ) and monomial realization M(λ). As an application, we also give an explicit form of monomial realization M(λ) in terms of Weyl groups.