Monomial Relization of Crystal Bases for Special Linear Lie Algebras

We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima. Introduction The quantum groups, which are certain deformations of the universal enveloping algebras of Kac-Moody algebras, were introduced independently by V. G. Drinfeld and M. Jimbo [1, 4]. In [6, 7], M. Kashiwara developed the crystal basis theory for integrable modules over quantum groups. 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. Moreover, crystal bases have a remarkably nice behavior with respect to taking the tensor product. In [13], while studying the structure of quiver varieties, H. Nakajima discovered that one can define a crystal structure on the set of irreducible components of a lagrangian subvariety Z of the quiver variety M. These irreducible components are identified with certain monomials, and the action of Kashiwara operators can be interpreted as multiplication by monomials. Moreover, in [13] and [8], M. Kashiwara and H. Nakajima gave a crystal structure on the set M of monomials and they showed that the connected component M(λ) of M containing a highest weight vector M with a dominant integral weight λ is isomorphic to the irreducible highest weight crystal B(λ). Therefore, a natural question arises: for each dominant integral weight λ, can we give an explicit characterization of the monomials in M(λ)? In this paper, for any dominant integral weight λ, we give an explicit description of the crystal M(λ) for special linear Lie algebras. In addition, we discuss the connection between the monomial realization and tableau realization of crystal bases given by Kashiwara and Nakashima. More precisely, let T (λ) denote the crystal consisting of semistandard tableaux This research was supported by KOSEF Grant # 98-0701-01-5-L and the Young Scientist Award, Korean Academy of Science and Technology. This research was supported by KOSEF Grant # 98-0701-01-5-L and BK21 Mathematical Sciences Division, Seoul National University. 1 2 SEOK-JIN KANG, JEONG-AH KIM AND DONG-UY SHIN of shape λ. Then we show that there exists a canonical crystal isomorphism between M(λ) and T (λ), which has a very natural interpretation in the language of insertion scheme. This work was initiated when the authors visited RIMS, Kyoto University, in the spring of 2002. We would like to express our sincere gratitude to Professor M. Kashiwara for his kindness and stimulating discussions during our visit. 1. Crystal bases Let I be a finite set and set A = (aij)i,j∈I be a generalized Cartan matrix. Consider the Cartan datum (A,Π,Π, P, P), where P = ( ⊕