Isotropic Grassmannians, Plücker and Cartan maps

This work is motivated by the relation between KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of sections dual determinantal Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate Cartan map, which, for a vector space V dimension N, embeds Grassmannian GrV0(V+V*) maximal isotropic subspaces + V*, with respect natural scalar product, into projectivization exterior Λ(V), Plücker which GrV(V V*) all N-planes in V* ΛN(V V*). The coordinates on are expressed bilinearly terms coordinates, holomorphic bundle Pf*→GrV0(V+V*,Q). affine big cell, this equivalent an identity Cauchy–Binet type, expressing determinants square submatrices skew symmetric N × matrix bilinear sums Pfaffians their principal minors.