Lagrangian Grassmannians, CKP Hierarchy and Hyperdeterminantal Relations

This work concerns the relation between the geometry of Lagrangian Grass-mannians 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 (R) V* into the projectivization of the exterior space AV is defined by restricting the Plucker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization V F* 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 x 2 x 2 quartic hyperdeterminantal relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with V (R) V* replaced by a polarized Hilbert space H = H+ (R) H-, with symplectic form w. The image of the Plucker map in the fermionic Fock space F = A(infinity/2)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 r-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.