论文标题
从$ b $ -toda到BKP层次结构
From the $B$-Toda to the BKP hierarchy
论文作者
论文摘要
结果表明，BKP层次结构的所有$τ$ - 函数都可以写成偏斜矩阵的pfaffians。 BKP层次结构的$τ$ - 函数由通用正交Grassmannian歧管（UOGM）中的点进行了参数。 UOGM是Schubert细胞的不相交联合，我们通过在SATO的意义上构造一个UOGM的框架来对每个Schubert细胞中的点进行分类并给出明确的参数化。然后，$τ$ - 功能以这些帧和Schur-Q函数表示。为了具体，我们对$ b $ -TODA的$τ$ - 功能进行了全面研究，可以将其视为BKP层次结构的有限版本。一路走来，我们还为复杂的纯纺纱器Du E. Cartan提供了建设性的描述。作为我们建筑的应用，我们由于A. Alexandrov而拒绝了Aorem，该定理指出，KDV将BKP求解为$ 2 $的时间参数恢复。我们通过证明KDV层次结构可以看作是BKP层次结构的$ 4 $。这种解释为BKP层次结构内的KDV轨道提供了完整的特征。除了代表理论的一些事实外，我们用来显示上述结果的主要工具是令人惊讶的简单线性代数。
It is shown that all $τ$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $τ$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM is a disjoint union of Schubert cells, we classify and give explicit parameterization for points in each Schubert cell by constructing a frame for UOGM in the sense of Sato. $τ$-functions are then expressed in terms of these frames and Schur-Q functions. For concreteness we give a comprehensive study for the $τ$-functions of $B$-Toda which can be viewed as a finite version of the BKP hierarchy. Along the way we also give a constructive description for complex pure spinors du E. Cartan. As an application of our construction, we reprove a theorem due to A. Alexandrov which states that KdV solves BKP up to rescaling of the time parameters by $2$. We prove this by showing that the KdV hierarchy can be viewed as $4$-reduction of the BKP hierarchy. This interpretation gives complete characterization for the KdV orbits inside the BKP hierarchy. Other than a few facts from representation theory, the main tools we use to show the above results, however, are surprisingly simple linear algebra.
