This work concerns the relation between the geometry of Lagrangian

Grassmannians and the CKP integrable hierarchy. The Lagrange map from the

Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite

dimensional symplectic vector space $V\oplus V^*$ into the projectivization of

the exterior space $\Lambda V$ is defined by restricting the Pl\”ucker map on

the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with

projection to the subspace of symmetric elements under dualization $V

\leftrightarrow V^*$. 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

\times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to

the CKP hierarchy, the Lagrangian Grassmannian framework is extended to

infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $

{\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form

$\omega$. The image of the Plucker map in the fermionic Fock space ${\mathcal

F}= \Lambda^{\infty/2}{\mathcal H}$ is identified 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

$\tau$-function at translates of a point in the space of odd flow variables

along the cubic lattices generated by power sums in the parameters.