Crystal Bases for Quantum Classical Algebras and Nakajima’s Monomials

Abstract

Using Nakajima's monomials, we construct a new realization of crystal bases for finite dimensional irreducible modules over quantum classical algebras. We also give an explicit bijection between the monomial realization and the Young tableau realization of crystal bases.

Level 0 Monomial Crystals

“…Our crystal structure described here is probably the same as one in [24] if we use the isomorphism between our B ,0,0 and Kashiwara-Nakashima's tableaux [23] in [15]. Note that the uniqueness of the crystal base of W ( ) was proved in [24].…”

“…The statement (d) is not explicitly stated in [15], but follows from [15,Prop. 3.2] or the argument below.…”

“…Finally we consider the case = n − 1 or n. Following [32], [15] we define the half size numbered box as…”

Monomial Realization of Crystal Bases B(∞) for the Quantum Finite Algebras

“…If i = n − 1 or n, thenf i M is obtained from M by multiplying A n−1 ( j ) −1 or A n ( j ) −1 , and a n−1 ( j ) + a n ( j ) = a n−2 ( j ) (1 ≤ j ≤ n − 2) in M. Note that y n−1 ( j ) = −a n−1 ( j − 1) + a n−2 ( j ) − a n−1 ( j ) = −a n−1 ( j − 1) + a n ( j ), y n ( j ) = −a n ( j − 1) + a n−2 ( j ) − a n ( j ) = −a n ( j − 1) + a n−1 ( j ), and by the definition of the Kashiwara operatorf i , they should be larger than 0. This contradicts condition (5). If i is neither n − 1 nor n, we use the same argument as for the C n type.…”

“…Moreover, they showed by means of mutually different ways that the connected component C(M) containing a maximal vector M of weight λ ∈ P + is isomorphic to the irreducible highest weight crystal B(λ). For the quantum algebras of type A n , B n , C n , D n , G 2 and A (1) n , Kang and the authors gave an explicit characterization of the Nakajima monomials in C(M) [4,5,12,21].…”