Nakajima Monomials and Crystals for Special Linear Lie Algebras

The theory of Nakajima monomials is a combinatorial scheme for realizing crystal bases of quantum groups. Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals in [16]. Kashiwara and Nakajima independently defined a crystal structure on the set of Nakajima monomials and also gave a realization of irreducible highest weight crystal B(λ) in terms of Nakajima monomials, as the connected component of the monomial set containing a maximal vector of dominant integral weight λ [9, 17]. This has lead to the belief that it should be possible to give a similar realization for B(∞), which is the crystal base of the negative part U q (g) of a quantum group over symmetrizable Kac-Moody algebra g, also. Much effort has been made [1, 5, 11, 13, 18–21] to give realization of B(∞) over various Kac-Moody algebras. In addition to these works, in our recent work [3, 14], we gave new realization of B(∞) for the finite simple Lie algebras, in terms of Young tableaux. Starting from the realization theorem of Kashiwara and Nakajima [9, 17], we can argue that it is not possible to find the crystal B(∞) within the set of Nakajima monomials with their given crystal structure. Hence, in our work [13], we constructed the set of extended Nakajima monomials and developed a crystal structure on it, and also gave explicit descriptions of B(∞) for A (1) n case, in the language of extended Nakajima monomials. 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 that on the extended monomial set. As the first contribution of our present paper, we introduce explicit descriptions of the crystal B(∞), in terms of extended Nakajima monomials. We restrict ourselves to special Linear Lie algebras. The extended Nakajima monomial description is obtained by relating it to the Young tableau realization [3]. The second contribution of this paper is to give an explicit description of the irreducible highest weight crystal B(λ) for any dominant integral weight λ, in monomial