Quantum superalgebras and the free-fermionic Yang-Baxter equation

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 U q ( gl ( 1 | 1 ) ) . We demonstrate that R -matrices from the finite quantum superalgebra U q ( 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 U q ( gl ˆ ( 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.