---
abstract: 'We study the crystal base of the negative part of a quantum group. Two explicit descriptions of the crystal $\B(\infty)$ for types $G_2$ are given. The first is given in terms of *extended Nakajima monomials* and the second realization follows a similar result given for other finite types by *Cliff*.'
address: |
    Korea Institute for Advanced Study\
    207-43 Cheongnyangni 2-dong, Dongdaemun-gu\
    Seoul 130-722, Korea
author:
- Hyeonmi Lee
title: 'Descriptions of the crystal $\B(\infty)$ for $G_2$'
---

[^1]

Introduction {#Introduction}
============

Quantum group $U_q(\mathfrak{g})$ is a $q$-deformation of the universal enveloping algebra over a Lie algebra $\mathfrak{g}$, and crystal bases reveal the structure of $U_q(\mathfrak{g})$-modules in a very simplified form. As these $U_q(\mathfrak{g})$-modules are known to be $q$-deformations of modules over the original Lie algebras, knowledge of these structures also affects the study of Lie algebras.

The crystal $\Binf$, which is the crystal base of the negative part $U_q^-(\mathfrak{g})$ of a quantum group, has received attention since the very birth of crystal base theory [@K91; @K90]. This is not only because it is an essential part of the *grand loop* argument proving the existence of crystal bases, but because it gives insight into the structure of quantum group 