The free-fermion point refers to a GL(2)×GL(1) parametrized Yang-Baxter equation within the six-vertex model. It has been known for a long time that this is connected with the quantum group Uq(gl(1|1)). We demonstrate that R-matrices from the finite quantum superalgebra Uq(gl(1|1)) produce a dense subset of the free-fermionic Yang-Baxter equations of the six-vertex model, matching those of the prime, simple modules in the affine quantum superalgebra Uq(glˆ(1|1)). Either of these quantum groups can be used to generate the full free-fermion point, and we discuss them both. Our discussion includes 6 families of six-vertex models used by Brubaker, Bump, and Friedberg in connection with Tokuyama's theorem, a deformation of the Weyl character formula. Thus our work gives quantum group interpretations for those models, known informally as Tokuyama ice.