Boundary polarization of the rational six-vertex model on a semi-infinite lattice

We consider the six-vertex model on a finite square lattice with the so-called partial domain wall boundary conditions. For the case of the rational Boltzmann weights, we compute the polarization on the free boundary of the lattice. For the finite lattice the result is given in terms of a ratio of determinants. In the limit where the side of the lattice with the free boundary tends to infinity (the limit of a semi-infinite lattice), they simplify and can be evaluated in a closed form.