Partial domain wall partition functions

We consider six-vertex model configurations on an (n × N) lattice,n ≤ N, that satisfy a variation on domain wall boundary conditions that we define and callpartial domain wall boundary conditions. We obtain two expressions for the correspondingpartial domain wall partition function, as an (N × N)-determinant and as an (n × n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KPτ-function, and, recalling that these determinants represent tree-level structure constants in N = 4\;SYM , we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.