Pfaffian

of a skew-symmetric matrix $X$

The polynomial $\text{Pf } X$ in the entries of $X$ whose square is $\text{det } X$. More precisely, if $X = \|x_{ij}\|$ is a skew-symmetric matrix (i.e. $x_{ij}=-x_{ji}$, $x_{ii}=0$; such a matrix is sometimes also called an alternating matrix) of order $2n$ over a commutative-associative ring $A$ with a unit, then $\text{Pf } X$ is the element of $A$ given by the formula

$$ \text{Pf } X = \sum_s \varepsilon(s)x_{i_1j_1}\ldots x_{i_nj_n}, $$

where the summation is over all possible partitions $s$ of the set $\{1,\ldots,2n\}$ into non-intersecting pairs $\{i_\alpha,j_\alpha\}$, where one may suppose that $i_\alpha<j_\alpha$, $\alpha=1,\ldots,n$, and where $\varepsilon(s)$ is the sign of the permutation

$$ \left( \begin{matrix} 1 & 2 & \ldots & 2n-1 & 2n \\ i_1 & j_1 & \ldots & i_n & j_n \end{matrix} \right). $$

A Pfaffian has the following properties:

1. $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;

1. $(\text{Pf } X)^2= \det X$;

3) if $E$ is a free $A$-module with basis $e_1,\ldots,e_{2n}$ and if

$$ u = \sum_{i < j} x_{ij} e_i \bigwedge e_j \in \bigwedge^2 A, $$

then

$$ \bigwedge^n u =n! (\text{Pf } X) e_1 \bigwedge \ldots \bigwedge e_{2n}. $$

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)