Isotropic Grassmannians, Plücker and Cartan maps

Abstract

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $\tau$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian ${\mathrm{Gr}}_V^0(V+V^{\ast })$ of maximal isotropic subspaces of $V+V^{\ast }$, with respect to the natural scalar product, into the projectivization of the exterior space $\Lambda (V)$, and the Plücker map, which embeds the Grassmannian ${\mathrm{Gr}}_V(V+V^{\ast })$ of all $N$-planes in $V+V^{\ast }$ into the projectivization of ${\Lambda }^N(V+V^{\ast })$. The Plücker coordinates on ${\mathrm{Gr}}_V^0(V+V^{\ast })$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${\mathrm{Pf}}^{\ast }\to {\mathrm{Gr}}_V^0(V+V^{\ast },Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric $N\times N$ matrix as bilinear sums over the Pfaffians of their principal minors.

References updated

References in corpus (1)

• On Addition Formulae of KP, mKP and BKP Hierarchies

Cited by in corpus (5)

• Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach
• Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations
• Tau functions, infinite Grassmannians and lattice recurrences
• Notes about KP/BKP correspondence
• Fredholm Pfaffian $\tau$-functions for orthogonal isospectral and isomonodromic systems