$R$ matrix for generalized quantum group of type $A$

The generalized quantum group $\mathcal{U}(ϵ)$ of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$. We prove that there exists a unique $R$ matrix on tensor product of fundamental type representations of $\mathcal{U}(ϵ)$ for arbitrary parameter sequence $ϵ$ corresponding to a non-conjugate Borel subalgebra of ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$. We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible $\mathcal{U}(ϵ)$-modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type $A_{M-1}^{(1)}$ or $A_{N-1}^{(1)}$.