itself.

Much effort has been made [@MR1475048; @ks; @saito; @MR1614241; @lee; @lee2; @HL] to give explicit description of the crystals $\Binf$ over various Kac-Moody algebras. A related known result is that it is possible to characterize the highest weight crystal $\B(\la)$ over symmetrizable Kac-Moody algebras, in terms of Nakajima monomials [@MR1988989; @MR2043368; @MR2074700; @kim; @shin], an object which was introduced by Nakajima [@MR1865400; @MR1988990]. This has lead to the belief that it should be possible to give a similar description for $\Binf$ also. Starting from a theorem of Kashiwara and Nakajima on the crystal structure of monomials [@MR1988989], we can argue that it is not possible to find the crystal $\B(\infty)$ within the set of Nakajima monomials with their given crystal structure. Hence, in our recent work [@lee], we constructed the set of extended monomials and developed a crystal structure on it, conjecturing that a certain connected component of the crystal would be isomorphic to $\B(\infty)$. Actually, the set of Nakajima monomials can be embedded as a subcrystal in this set of extended Nakajima monomials. Thus, the monomial theory developed for irreducible highest weight crystal can easily be transferred to the extended monomial set.

In the current work, we restrict 