Tau functions, infinite Grassmannians, and lattice recurrences

The addition formulae for KP τ-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the τ-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon–Pemantle) recurrence.