Generalized permutation matrices and non-weight modules over $\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }1)$

We study the category ${\mathcal{M}}_{\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }1)}(k\mathrm{\mid }k)$ of $\mathcal{U}(\mathfrak{h})\text{-free}$ $\mathcal{U}(\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }1))$-modules of rank $k$ in each parity (rank $(k\mathrm{\mid }k)$), where $k\in {\mathbb{Z}}_{\ge 1}$. We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.