ourselves to the $G_2$-type finite simple Lie algebra. For this case, we give an explicit set of extended Nakajima monomials and show it to be isomorphic to $\Binf$. The previous work [@HL], giving a Young tableaux realization of $\Binf$ for the finite simple types, is used in doing this. We also extend Cliff’s [@MR1614241] realization of $\Binf$ for classical finite types, given in terms of a completely different object, to the $G_2$-type.

The paper is organized as follows. We start by reviewing the notion of extended Nakajima monomials and the crystal structure given on the set of such monomials. Also, we cite Young tableau expression of crystal $\B(\infty)$ for type $G_2$ which play a crucial role in our work. We then proceed to give a monomial realization of the crystal $\B(\infty)$. In the process of obtaining these results, we give new expressions for the Kashiwara operators acting on the extended Nakajima monomials, more appropriate for the situation in hand. In the last section, we deal with Cliff’s approach of realizing $\Binf$.

Extended Nakajima monomials and Young tableaux {#Nakajima's monomials and crystals}
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In this section, we introduce notation and cite facts that are crucial for our work. Please refer to 