This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. For a complex vector space V of dimension N , and H N := V ⊕ V * the associated symplectic space, with canonical symplectic structure ω N , the exterior space Λ(H N ) is decomposed into a direct sum of irreducible Sp(H N , ω N ) submodules, and a basis adapted to this decomposition is constructed. The Lagrangian map) is defined by restricting the Plücker map to the Lagrangian Grassmannian Gr L V (H N , ω N ) of maximal isotropic subspaces and composing it with projection to the subspace of symmetric elements of Λ N (H N ) under dualization. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the 2 × 2 × 2 quartic hyperdeterminantal relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with H N replaced by a polarized Hilbert space H = H + ⊕ H − , with symplectic form ω. The fermionic Fock space F = Λ ∞/2 H is decomposed into a direct sum of irreducible Sp(H, ω) representations and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the τ -function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

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.