User:Rafael.greenblatt/sandbox/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

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

where the summation is over all possible partitions of the set into non-intersecting pairs , where one may suppose that , , and where is the sign of the permutation

A Pfaffian has the following properties:

1) for any matrix of order ;

2) ;

3) if is a free -module with basis and if

then

References

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