Lagrangian Grassmannians, CKP Hierarchy and Hyperdeterminantal Relations

This work concerns the relation between geometry of Lagrangian Grassmannians and CKP integrable hierarchy. The Lagrange map from Grassmannian maximal isotropic (Lagrangian) subspaces a finite dimensional symplectic vector space $V\oplus V^*$ into projectivization exterior $\Lambda V$ is defined by restricting Pl\"ucker on full to sub-Grassmannian composing it with projection subspace symmetric elements under dualization $V \leftrightarrow V^*$. In terms affine coordinate matrix big cell, this reduces principal minors map, whose image cut out $2 \times 2 2$ quartic {\em hyperdeterminantal} relations. To apply hierarchy, framework extended infinite dimensions, replaced polarized Hilbert $ {\mathcal H} ={\mathcal H}_+\oplus H}_-$, form $\omega$. Plucker in fermionic Fock ${\mathcal F}= \Lambda^{\infty/2}{\mathcal H}$ identified defined. linear constraints defining reduction hierarchy are expressed as null condition analogue hyperdeterminantal relations deduced. A multiparametric family such shown be satisfied evaluation $\tau$-function at translates point odd flow variables along cubic lattices generated power sums parameters.