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$$
 
$$
  
where the  summation is over all possible partitions <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250013.png" /> of the  set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250014.png" /> into  non-intersecting pairs <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250015.png" />, where one may  suppose that <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250016.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250017.png" />, and where <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250018.png" /> is the sign of  the permutation
+
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
  
<table class="eq"  style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250019.png"  /></td> </tr></table>
+
$$
 +
\left(
 +
\begin{matrix}{ccccc}
 +
1 & 2 & \ldots & 2n-1 & 2n \\
 +
i_1 & j_1 & \ldots & i_n & j_n
 +
\end{matrix}
 +
\right).
 +
$$
  
 
A Pfaffian has the following properties:
 
A Pfaffian has the following properties:

Revision as of 14:15, 25 January 2012

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

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

A Pfaffian has the following properties:

1) 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)
How to Cite This Entry:
Rafael.greenblatt/sandbox/Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rafael.greenblatt/sandbox/Pfaffian&oldid=20486