Crystal Bases and Monomials for Uq(G2)-modules

In this paper, we give a new realization of crystal bases for irreducible highest weight modules over Uq(G2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Introduction In 1985, the quantum groups Uq(g), which may be thought of as q-deformations of the universal enveloping algebras U(g) of Kac-Moody algebras g, were introduced independently by Drinfel’d and Jimbo [1, 4]. The integrable highest weight representations over symmetrizable Kac-Moody algebras can be deformed consistently to the highest weight representations over the corresponding quantum groups for generic q [20]. From this point of view, the crystal basis theory for integrable modules over quantum groups was developed by Kashiwara [10, 11]. Crystal bases can be viewed as bases at q = 0 and they are given a structure of colored oriented graphs, called the crystal graphs. Crystal graphs have many nice combinatorial properties reflecting the internal structure of integrable modules. In [13], Kashiwara and Nakashima gave an explicit realization of crystal bases of finite dimensional irreducible modules over classical Lie algebras using semistandard tableaux with given shapes satisfying certain additional conditions. Motivated by their work, Kang and Misra discovered a tableau realization of crystal bases for finite dimensional irreducible modules over the exceptional Lie algebra G2 [9]. In [17], Littelmann gave another description of crystal bases for finite dimensional simple Lie algebras using the Lakshmibai-Seshadri monomial theory. His approach was generalized to the path model theory for all symmetrizable Kac-Moody algebras [18, 19]. ∗This research was supported by KOSEF Grant # 98-0701-01-5-L