# Pfaffians

Contents

## Pfaffians

### 线性空间上的反对称算子

\begin{align} \langle\cdot,\cdot\rangle:V\otimes V&\to K\\ \mathbf{e}_i\otimes\mathbf{e}_j&\mapsto \delta_{ij}. \end{align}

$$\langle\mathbf{u},\hat{A}\mathbf{u}\rangle=0,\quad\forall\mathbf{u}\in V.$$

\begin{align} \varphi:\wedge^2 V&\to\text{Asym}(V)\\ \mathbf{a}\wedge\mathbf{b}&\mapsto [\mathbf{x}\mapsto\mathbf{a}\langle\mathbf{b},\mathbf{x}\rangle-\mathbf{b}\langle\mathbf{a},\mathbf{x}\rangle]. \end{align}

$$\text{Pf}(\hat{A})\mathbf{e}_1\wedge\dotsb\wedge\mathbf{e}_N=\frac{1}{(N/2)!}\bigwedge_{k=1}^{N/2}\alpha,$$

$$\det(\hat{A})=\text{Pf}(\hat{A})^2.$$

$$\alpha=\mathbf{v}_1\wedge\mathbf{v}_2+\dotsb+\mathbf{v}_{k-1}\wedge\mathbf{v}_ k,$$

$$\mathbf{v}_1\wedge\dotsb\wedge\mathbf{v}_N=\text{Pf}(\hat{A})\mathbf{e}_1\wedge\dotsb\wedge\mathbf{e}_ N.$$

$$\mathbf{u}_1\wedge\dotsb\wedge\mathbf{u}_N=\text{Pf}(\hat{A})^{-1}\mathbf{e}_1\wedge\dotsb\wedge\mathbf{e}_ N,$$

$$\det(\hat{A})\mathbf{u}_1\wedge\dotsb\wedge\mathbf{u}_N=\hat{A}\mathbf{u}_1\wedge\dotsb\wedge\hat{A}\mathbf{u}_N=\mathbf{v}_1\wedge\dotsb\wedge\mathbf{v}_ N.$$

$$\det(\hat{A})=\text{Pf}(\hat{A})^2.$$

### 域上的反对称矩阵

$$a_{ii}=0,\quad a_{ij}+a_{ji}=0,\quad\forall i,j\in\{1,\dotsc,N\}.$$

\begin{align} \text{Mat}_ N(K)&\to\text{End}(V)\\ A=(a_{ij})&\mapsto\hat{A}=[\mathbf{e}_j\mapsto\sum_ia_{ij}\mathbf{e}_ i] \end{align}

$$\alpha=\varphi^{-1}(\hat{A})=\sum_{i < j}a_{ij}\mathbf{e}_ i\wedge\mathbf{e}_ j,$$ 由Pfaffian的定义, 以及定理1.7, 得到

### 交换幺环上的反对称矩阵

$$a_{ii}=0,\quad a_{ij}+a_{ji}=0,\quad\forall i,j\in\{1,\dotsc,N\}.$$

$$\det(A)=\text{Pf}(A)^2.$$

\begin{align} \varphi:S&\to R\\ x_{ij}&\mapsto a_{ij}, \end{align}

$$\det(A)=\psi(\det(B))=\psi(\text{Pf(B)}^2)=\text{Pf(A)}^2.$$