Automatic Construction of Explicit R Matrices for the One-Parameter Families of Irreducible Typical Highest Weight (0|α) Representations of U_q[gl(m|n)]

We detail the automatic construction of R matrices corresponding to (the tensor products of) the (0|\alpha) families of highest-weight representations of the quantum superalgebras U_q[gl(m|n)]. These representations are irreducible, contain a free complex parameter \alpha, and are 2^{mn} dimensional. Our R matrices are actually (sparse) rank 4 tensors, containing a total of 2^{4mn} components, each of which is in general an algebraic expression in the two complex variables q and \alpha. Although the constructions are straightforward, we describe them in full here, to fill a perceived gap in the literature. As the algorithms are generally impracticable for manual calculation, we have implemented the entire process in Mathematica; illustrating our results with U_q[gl(3|1)].