Extended Nakajima monomials and crystal [formula omitted] for finite simple Lie algebras

Abstract

Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals B ( λ ) . The monomial set can be extended so that it contains the crystal B ( ∞ ) in addition to B ( λ ) . We study the crystal base B ( ∞ ) of the negative part of a quantum group. For finite types B n , C n , D n , and G 2 , we present new descriptions of the crystal B ( ∞ ) in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau description, which is another description of B ( ∞ ) .