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The sign convention for the Pfaffian of a skew-symmetric matrix on the Wikipedia looks like the negative of the sign convention in Lang's Algebra (at least looking at the 4 by 4 matrix in the Examples section):

Obviously if we were starting mathematics from scratch there would be no harm in defining it either way, but I wonder if one way is more standard. (In the application I have in mind, Lang's way is better.)

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up vote 2 down vote accepted

Whenever you want to regard a Pfaffian as a sum over matchings (just as a determinant is a sum over permutations), it is standard to assign the sign plus to the term $a_{12}a_{34}\dots a_{2n-1,2n}$ and I almost never see another convention, so I would say that this is standard and it has several advantages.

Clearly, if your matrices happen to be block matrices, the convention $Pf \left( \begin{smallmatrix} 0& I \\ -I& 0 \end{smallmatrix}\right) =1$ is more comfortable, but there is the very simple solution of just defining $\widetilde{Pf} $ with the appropriate sign change at the beginning of your paper or at least simply state that there are two different conventions and which one you use and how it relates to the other one (this is at most two lines).

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