论文标题
各向同性司法，普吕克和卡坦地图
Isotropic Grassmannians, Plücker and Cartan maps
论文作者
论文摘要
这项工作是由KP和BKP集成层次结构之间的关系的动机，他们的$τ$ - 功能可以看作是无限尺寸格拉曼尼亚人的双重决定性和PFAFFIAN线捆绑的部分。在有限尺寸中，我们展示了如何关联卡坦地图，对于矢量空间$ v $ dimension $ n $，嵌入了Grassmannian $ {\ mathrm {gr}}^0_v（v+ v^*）$最大同位素的基础$ v+ v^*$的最大spacior $ $ v^*$ $ $ v^*和Plücker地图，该地图将所有$ n $ - 平面的Grassmannian $ {\ Mathrm {gr}} _ V（V+ V^*）$嵌入$ V+ V^*$中的所有$ n $ planes $。 plücker在$ {\ mathrm {gr}}^0_v（v+v^*）$上的坐标是用cartan坐标表示双线的，这是双pfaffian线bundle $ { {gr}}^0_v（v+v^*，q）$。就大细胞上的仿射坐标而言，这等同于cauchy-binet类型的身份，表达了偏斜的对称$ n \ times n $矩阵的正方形子膜的决定因素，就像其主要少年人的pfaffians上。
This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $τ$-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 $Λ(V)$, and the Plücker map, which embeds the Grassmannian ${\mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $Λ^N(V + V^*)$. The Plücker 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.
