Young tableaux and crystal for the exceptional Lie algebra types

We study the crystal base B ( ∞ ) associated with the negative part of the quantum group for finite simple Lie algebras of types E 6 , E 7 , E 8 , and F 4 . After describing the crystal B ( ∞ ) as a union of highest weight crystals, we give an explicit description of B ( ∞ ) in terms of Young tableaux.