Pfaffian

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

The Pfaffian (of a skew-symmetric matrix $X$) is the polynomial $\def\Pf{\mathrm{Pf}\;} \Pf X$ in the entries of $X$ whose square is the determinant $\det X$. 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 $\Pf X$ is the element of $A$ given by the formula

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

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 $\epsilon(s)$ 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. $\Pf (C^T X C) = (\det C) (\Pf X)$ for any matrix $C$ of order $2n$;
2. $(\Pf X)^2= \det X$;
3. if $E$ is a free $A$-module with basis $e_1,\ldots,e_{2n}$ and if $$ u = \sum_{i < j} x_{ij} e_i \wedge e_j \in \bigwedge^2 A, $$ then $$ \bigwedge^n u =n! (\Pf X) e_1 \wedge \ldots \wedge e_{2n}. $$

References

[Bo]	N. Bourbaki, "Elements of mathematics", 2. Linear and multilinear algebra, Addison-Wesley (1973) pp. Chapt. 2 (Translated from French) MR0274237