Crystal base of the negative half of the quantum superalgebra $U_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n))$

We construct a crystal base of $U_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n){)}^{-}$, the negative half of the quantum superalgebra $U_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n))$. We give a combinatorial description of the associated crystal ${\mathcal{B}}_{m\mathrm{\mid }n}(\mathrm{\infty })$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(\lambda )$. We also construct a crystal base of a parabolic Verma module $X(\lambda )$ associated with the subalgebra $U_q({\mathfrak{g}\mathfrak{l}}_{0\mathrm{\mid }n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n){)}^{-}$ and the Kac module $K(\lambda )$ under the canonical embedding and projection of $X(\lambda )$ to $U_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n){)}^{-}$ and $K(\lambda )$, respectively.