This work is motivated by the relation between the KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of 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 ${\mathrm{G}\mathrm{r}}_V^0(V+V^{*})$ of maximal isotropic subspaces of V + V*, with respect to the natural scalar product, into the projectivization of the exterior space Λ(V), and the Plücker map, which embeds the Grassmannian Gr_V(V + V*) of all N-planes in V + V* into the projectivization of Λ^N(V + V*). The Plücker coordinates on ${\mathrm{G}\mathrm{r}}_V^0(V+V^{*})$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${\mathrm{P}\mathrm{f}}^{*}\to {\mathrm{G}\mathrm{r}}_V^0(V+V^{*},Q)$. 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 × N matrix as bilinear sums over the Pfaffians of their principal minors.

中文翻译：

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各向同性的格拉斯曼人，普吕克和卡丹地图

这项工作是由KP和BKP可积层次之间的关系所推动的，它们的τ函数可以看作是无限维草曼上的双行列式和Pfaffian线束的截面流动。在有限维度上，我们展示了如何关联Cartan映射，该映射对于维度为N的向量空间V嵌入了格拉斯曼矩阵${\mathrm{G}\mathrm{[R}}_{\mathrm{伏特}}^0（\mathrm{伏特}+{\mathrm{伏特}}^{*}）$V + V *的最大各向同性子空间（相对于自然标量积）到外部空间Λ（V）的投影中，以及Plücker映射，该图嵌入了所有N的Grassmannian Gr _V（V + V *）-平面在V + V *成Λ的projectivization ^ñ（V + V *）。普吕克（Plücker）协调${\mathrm{G}\mathrm{[R}}_{\mathrm{伏特}}^0（\mathrm{伏特}+{\mathrm{伏特}}^{*}）$用Cartan坐标双线性表示，这是双Pfaffian线束的全纯截面。${\mathrm{P}\mathrm{F}}^{*}\to {\mathrm{G}\mathrm{[R}}_{\mathrm{伏特}}^0（\mathrm{伏特}+{\mathrm{伏特}}^{*}，问）$。就大单元上的仿射坐标而言，这等同于柯西-比涅特类型的恒等式，将偏向对称N × N矩阵的平方子矩阵的行列式表示为其主要未成年人的Pfaffian上的双线性和。