Isotropic Grassmannians, Pl\"ucker and Cartan maps

TL;DR

This paper explores the geometric relations between isotropic Grassmannians, Pl"ucker, and Cartan maps, revealing how these embeddings relate through bilinear identities involving determinants and Pfaffians, with implications for integrable hierarchies.

Contribution

It establishes explicit relations between Cartan and Pl"ucker maps for isotropic Grassmannians, expressing Pl"ucker coordinates in terms of Cartan coordinates via bilinear identities.

Findings

Expressed Pl"ucker coordinates bilinearly in terms of Cartan coordinates.

Derived identities relating determinants of submatrices to Pfaffians of minors.

Connected geometric embeddings to identities relevant for integrable hierarchies.

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 ${\mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $\Lambda(V)$, and the Pl\"ucker map, which embeds the Grassmannian ${\mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $\Lambda^N(V + V^*)$. The Pl\"ucker coordinates on ${\mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${\mathrm {Pf}}^* \rightarrow {\mathrm {Gr}}^0_V(V+V^*, Q)$. 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.